title

Logs (logarithms), Clearly Explained!!!

description

Here's a little video to help you bone up on logs. Logs are used all over statistics, so it's good to have a solid understanding of them.
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detail

{'title': 'Logs (logarithms), Clearly Explained!!!', 'heatmap': [{'end': 338.497, 'start': 293.599, 'weight': 0.709}, {'end': 508.261, 'start': 487.934, 'weight': 0.723}, {'end': 563.289, 'start': 509.442, 'weight': 0.742}], 'summary': 'Provides a clear explanation of logarithms, focusing on log base 2 scale, demonstrating how numbers can be rewritten as powers of 2 and the use of log scale for plotting fold change. it also explains the use of logbase 2 in qpcr to interpret data and calculate the geometric mean, highlighting the benefits of using the geometric mean for log-based data.', 'chapters': [{'end': 445.287, 'segs': [{'end': 158.686, 'src': 'embed', 'start': 118.931, 'weight': 0, 'content': [{'end': 129.517, 'text': 'The log function just isolates the exponent, and that means that 3 is the log base 2 equivalent of 8 on our log base 2 axis.', 'start': 118.931, 'duration': 10.586}, {'end': 134.879, 'text': "Now let's take the log base 2 of 4.", 'start': 131.037, 'duration': 3.842}, {'end': 141.84, 'text': 'First we rewrite 4 as a power of 2, and the log function just isolates the exponent.', 'start': 134.879, 'duration': 6.961}, {'end': 158.686, 'text': 'Since 2 is the exponent, the log base 2 of 2 to the 2 equals 2, and thus 2 is the log base 2 equivalent of the number 4 on our log axis.', 'start': 143.34, 'duration': 15.346}], 'summary': 'Log base 2 of 8 is 3, and log base 2 of 4 is 2.', 'duration': 39.755, 'max_score': 118.931, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VSi0Z04fWj0/pics/VSi0Z04fWj0118931.jpg'}, {'end': 256.68, 'src': 'embed', 'start': 229.23, 'weight': 2, 'content': [{'end': 234.411, 'text': 'Guess what the log function is just about to do? Isolate the exponent.', 'start': 229.23, 'duration': 5.181}, {'end': 236.292, 'text': 'Boring, we already know about this.', 'start': 234.791, 'duration': 1.501}, {'end': 239.112, 'text': 'We know that the log function isolates the exponent.', 'start': 236.612, 'duration': 2.5}, {'end': 244.773, 'text': 'Bam! Negative 1 is the log base 2 of 1 half.', 'start': 239.932, 'duration': 4.841}, {'end': 254.339, 'text': "Okay, now we've got 1 over 4 on our number line, and 1 fourth can be rewritten as a power of 2.", 'start': 246.193, 'duration': 8.146}, {'end': 256.68, 'text': '2 to the negative 2.', 'start': 254.339, 'duration': 2.341}], 'summary': 'The log function isolates exponents, e.g., log base 2 of 1/2 is -1.', 'duration': 27.45, 'max_score': 229.23, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VSi0Z04fWj0/pics/VSi0Z04fWj0229230.jpg'}, {'end': 338.497, 'src': 'heatmap', 'start': 293.599, 'weight': 0.709, 'content': [{'end': 300.102, 'text': 'and negative 3 then is the equivalent on the log 2 scale of 1 over 8..', 'start': 293.599, 'duration': 6.503}, {'end': 308.666, 'text': 'Notice how much space is between 1 and 8, and how little space is between 1 and 1 over 8.', 'start': 300.102, 'duration': 8.564}, {'end': 318.487, 'text': 'A measurement way up here is 8 times greater than 1, and a measurement down here is 8 times less than 1.', 'start': 308.666, 'duration': 9.821}, {'end': 327.211, 'text': 'Even though both measurements represent the same magnitude in fold change relative to 1, the distance from 1 is not symmetric.', 'start': 318.487, 'duration': 8.724}, {'end': 333.154, 'text': 'In contrast, the magnitude is equidistant on the log axis.', 'start': 328.432, 'duration': 4.722}, {'end': 338.497, 'text': 'This is why fold changes should always be plotted on log axes.', 'start': 334.235, 'duration': 4.262}], 'summary': 'Log 2 scale shows asymmetry in fold change, favoring log axis for plotting fold changes.', 'duration': 44.898, 'max_score': 293.599, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VSi0Z04fWj0/pics/VSi0Z04fWj0293599.jpg'}, {'end': 377.311, 'src': 'embed', 'start': 346.455, 'weight': 1, 'content': [{'end': 355.38, 'text': 'For example, the log base 2 of 8 equals the log base 2 of 2 to the third, and if you isolate the exponent, you get 3.', 'start': 346.455, 'duration': 8.925}, {'end': 363.704, 'text': 'The other part of the take-home message is that you should use a log scale or axis when talking about fold change.', 'start': 355.38, 'duration': 8.324}, {'end': 368.107, 'text': 'This puts positive and negative fold changes on a symmetric scale.', 'start': 364.265, 'duration': 3.842}, {'end': 375.01, 'text': '8 fold up is the same distance from 0 as 8 fold down.', 'start': 370.146, 'duration': 4.864}, {'end': 377.311, 'text': "Here's the weird thing.", 'start': 376.331, 'duration': 0.98}], 'summary': 'Using log scale for fold change provides symmetric representation.', 'duration': 30.856, 'max_score': 346.455, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VSi0Z04fWj0/pics/VSi0Z04fWj0346455.jpg'}, {'end': 445.287, 'src': 'embed', 'start': 415.031, 'weight': 4, 'content': [{'end': 420.892, 'text': 'Just so you know, R, a programming language that a lot of people use to do stats,', 'start': 415.031, 'duration': 5.861}, {'end': 429.574, 'text': "and I'd be willing to guess that most other programming languages do the same thing defines log base 2 of 0 to be equal to negative infinity.", 'start': 420.892, 'duration': 8.682}, {'end': 445.287, 'text': 'Intuitively. this makes sense because 1 over 2 to infinity equals the smallest number you can imagine, and in my mind, the smallest number I can imagine is 0..', 'start': 430.915, 'duration': 14.372}], 'summary': 'R defines log base 2 of 0 as negative infinity, making intuitive sense.', 'duration': 30.256, 'max_score': 415.031, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VSi0Z04fWj0/pics/VSi0Z04fWj0415031.jpg'}], 'start': 18.32, 'title': 'Logarithms and log base 2', 'summary': 'Explains the concept of logarithms, with a focus on the log base 2 scale, demonstrating how numbers can be rewritten as powers of 2 and how the log function isolates the exponent to determine the log base 2 equivalents of various numbers. it also emphasizes the importance of using log scale for plotting fold change to ensure symmetric representation and isolation of exponents, with examples of log base 2 of 8 equaling 3 and the absence of a value for log base 2 of 0, which is defined as negative infinity in r programming.', 'chapters': [{'end': 318.487, 'start': 18.32, 'title': 'Understanding logarithms and log base 2', 'summary': 'Explains the concept of logarithms and specifically focuses on the log base 2 scale, demonstrating how numbers can be rewritten as powers of 2, and how the log function isolates the exponent to determine the log base 2 equivalents of various numbers on the scale.', 'duration': 300.167, 'highlights': ['The log base 2 scale is explained by rewriting numbers as powers of 2 and isolating the exponent using the log function, for instance, log base 2 of 8 equals 3, log base 2 of 4 equals 2, log base 2 of 2 equals 1, and log base 2 of 1 equals 0.', 'Demonstration of how the log function isolates the exponent for numbers less than 1, such as log base 2 of 1 half equals -1, log base 2 of 1 quarter equals -2, and log base 2 of 1 over 8 equals -3, highlighting the exponential difference between these numbers.']}, {'end': 445.287, 'start': 318.487, 'title': 'Importance of logarithmic scale for fold change', 'summary': 'Emphasizes the importance of using log scale for plotting fold change to ensure symmetric representation and isolation of exponents, with the example of log base 2 of 8 equals to 3 and the absence of a value for log base 2 of 0, which is defined as negative infinity in r programming.', 'duration': 126.8, 'highlights': ['Logs isolate exponents, demonstrated by the example of log base 2 of 8 equals the log base 2 of 2 to the third, resulting in 3.', 'Using a log scale ensures symmetric representation of positive and negative fold changes, where 8 fold up is equidistant from 0 as 8 fold down.', 'In R programming, log base 2 of 0 is defined as negative infinity, aligning with the intuition that 1 over 2 to infinity equals the smallest imaginable number, which is 0.']}], 'duration': 426.967, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VSi0Z04fWj0/pics/VSi0Z04fWj018320.jpg', 'highlights': ['The log base 2 scale is explained by rewriting numbers as powers of 2 and isolating the exponent using the log function, for instance, log base 2 of 8 equals 3, log base 2 of 4 equals 2, log base 2 of 2 equals 1, and log base 2 of 1 equals 0.', 'Using a log scale ensures symmetric representation of positive and negative fold changes, where 8 fold up is equidistant from 0 as 8 fold down.', 'Demonstration of how the log function isolates the exponent for numbers less than 1, such as log base 2 of 1 half equals -1, log base 2 of 1 quarter equals -2, and log base 2 of 1 over 8 equals -3, highlighting the exponential difference between these numbers.', 'Logs isolate exponents, demonstrated by the example of log base 2 of 8 equals the log base 2 of 2 to the third, resulting in 3.', 'In R programming, log base 2 of 0 is defined as negative infinity, aligning with the intuition that 1 over 2 to infinity equals the smallest imaginable number, which is 0.']}, {'end': 609.943, 'segs': [{'end': 473.696, 'src': 'embed', 'start': 445.287, 'weight': 0, 'content': [{'end': 448.868, 'text': "Here's an example of when using LogBase 2 is super cool.", 'start': 445.287, 'duration': 3.581}, {'end': 452.989, 'text': 'QPCR or real-time PCR.', 'start': 450.169, 'duration': 2.82}, {'end': 457.131, 'text': "If you don't know what PCR is, don't worry about it.", 'start': 454.49, 'duration': 2.641}, {'end': 464.153, 'text': "Just imagine that it's a process where after each step, whatever it's measuring doubles.", 'start': 457.711, 'duration': 6.442}, {'end': 473.696, 'text': 'The LogBase 2 makes sense with this data because each time the machine goes through a cycle, the number of PCR products doubles.', 'start': 465.533, 'duration': 8.163}], 'summary': 'Using logbase 2 in qpcr allows for measuring doubling of pcr products.', 'duration': 28.409, 'max_score': 445.287, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VSi0Z04fWj0/pics/VSi0Z04fWj0445287.jpg'}, {'end': 516.866, 'src': 'heatmap', 'start': 487.934, 'weight': 0.723, 'content': [{'end': 494.278, 'text': 'This is because the log base 2 of 2 to the 0 equals 0.', 'start': 487.934, 'duration': 6.344}, {'end': 500.681, 'text': 'The second time we do the qPCR, the machine says there are twice as many transcripts as the first time.', 'start': 494.278, 'duration': 6.403}, {'end': 508.261, 'text': 'In other words, the difference between the first and second runs was one cycle in the machine.', 'start': 501.997, 'duration': 6.264}, {'end': 516.866, 'text': 'The third time we do the qPCR, the machine says there are eight times as many transcripts as the first time.', 'start': 509.442, 'duration': 7.424}], 'summary': 'Qpcr shows exponential increase in transcripts over cycles.', 'duration': 28.932, 'max_score': 487.934, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VSi0Z04fWj0/pics/VSi0Z04fWj0487934.jpg'}, {'end': 577.661, 'src': 'heatmap', 'start': 509.442, 'weight': 1, 'content': [{'end': 516.866, 'text': 'The third time we do the qPCR, the machine says there are eight times as many transcripts as the first time.', 'start': 509.442, 'duration': 7.424}, {'end': 523.909, 'text': 'In other words, there was a difference of three cycles between the first run and the last run.', 'start': 518.067, 'duration': 5.842}, {'end': 526.932, 'text': "Now, here's a question.", 'start': 525.231, 'duration': 1.701}, {'end': 535.885, 'text': "What's the average of the three runs? The average is 3.7 if we look at the quantities of transcripts.", 'start': 527.913, 'duration': 7.972}, {'end': 541.488, 'text': 'The average is 1.3 if we look at the differences in cycles.', 'start': 537.206, 'duration': 4.282}, {'end': 554.502, 'text': 'If we raise 2 by 1.3, the mean of the logs, to convert back to normal numbers, we get 2 to the 1.3 equals 2.5.', 'start': 542.729, 'duration': 11.773}, {'end': 563.289, 'text': "The mean calculated with the logs is less than the one calculated with normal numbers, because it wasn't as swayed by the third run,", 'start': 554.502, 'duration': 8.787}, {'end': 565.651, 'text': 'when we got eight times as many transcripts.', 'start': 563.289, 'duration': 2.362}, {'end': 573.517, 'text': 'That third run is a bit of an outlier and the mean of the logs, called the geometric mean,', 'start': 566.712, 'duration': 6.805}, {'end': 577.661, 'text': 'is more robust to the effects of outliers than the mean of normal numbers.', 'start': 573.517, 'duration': 4.144}], 'summary': 'Three qpcr runs show 8x increase in transcripts, averaging 3.7 in quantity and 1.3 in cycles, indicating outlier impact on mean.', 'duration': 23.159, 'max_score': 509.442, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VSi0Z04fWj0/pics/VSi0Z04fWj0509442.jpg'}], 'start': 445.287, 'title': 'Using logbase 2 in qpcr and mean calculation', 'summary': 'Explains the use of logbase 2 in qpcr to interpret data and calculate the geometric mean, noting that every cycle doubles the amount of transcripts, and demonstrates how the mean calculated with logs is less affected by outliers than the mean of normal numbers.', 'chapters': [{'end': 609.943, 'start': 445.287, 'title': 'Logbase 2 in qpcr and mean calculation', 'summary': 'Explains the use of logbase 2 in qpcr to interpret data and calculate the geometric mean, noting that every cycle doubles the amount of transcripts, and demonstrates how the mean calculated with logs is less affected by outliers than the mean of normal numbers.', 'duration': 164.656, 'highlights': ['The use of LogBase 2 in qPCR is explained, emphasizing that every cycle doubles the amount of transcripts, which can lead to outliers.', 'The demonstration of calculating the geometric mean with logs and its robustness to the effects of outliers compared to the mean of normal numbers.', 'The explanation of how log base 2 makes sense in qPCR data interpretation, with examples of differences in cycles and average calculation.']}], 'duration': 164.656, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VSi0Z04fWj0/pics/VSi0Z04fWj0445287.jpg', 'highlights': ['The use of LogBase 2 in qPCR is explained, emphasizing that every cycle doubles the amount of transcripts, which can lead to outliers.', 'The demonstration of calculating the geometric mean with logs and its robustness to the effects of outliers compared to the mean of normal numbers.', 'The explanation of how log base 2 makes sense in qPCR data interpretation, with examples of differences in cycles and average calculation.']}, {'end': 936.255, 'segs': [{'end': 702.694, 'src': 'embed', 'start': 672.214, 'weight': 2, 'content': [{'end': 676.174, 'text': 'Second, we rewrite the numbers as powers of two.', 'start': 672.214, 'duration': 3.96}, {'end': 681.575, 'text': 'Just like a normal multiplication, we add the exponents together.', 'start': 677.414, 'duration': 4.161}, {'end': 685.556, 'text': 'The log function just isolates the exponents.', 'start': 682.435, 'duration': 3.121}, {'end': 693.069, 'text': 'This shows that the log of multiplied numbers is just the sum of their exponents.', 'start': 686.926, 'duration': 6.143}, {'end': 698.052, 'text': "Now let's look at numbers that are not power of two friendly.", 'start': 694.95, 'duration': 3.102}, {'end': 702.694, 'text': 'First, wrap everything up in log base two functions.', 'start': 699.172, 'duration': 3.522}], 'summary': 'Rewriting numbers as powers of two, log isolates exponents, log of multiplied numbers is the sum of exponents, wrapping up in log base two functions.', 'duration': 30.48, 'max_score': 672.214, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VSi0Z04fWj0/pics/VSi0Z04fWj0672214.jpg'}, {'end': 863.127, 'src': 'embed', 'start': 804.858, 'weight': 0, 'content': [{'end': 810.36, 'text': 'This is because 8 fold up is the same distance from 0 as 8 fold down.', 'start': 804.858, 'duration': 5.502}, {'end': 811.92, 'text': "Everything's symmetrical.", 'start': 810.82, 'duration': 1.1}, {'end': 824.444, 'text': 'The third thing we learned is that the mean of logs, aka the geometric mean, is great for log-based data when something is doubling every time,', 'start': 814.221, 'duration': 10.223}, {'end': 825.004, 'text': 'for example.', 'start': 824.444, 'duration': 0.56}, {'end': 829.122, 'text': 'and is less sensitive to outliers than the normal mean.', 'start': 826.141, 'duration': 2.981}, {'end': 837.006, 'text': 'The fourth thing we learned is that the log of multiplication is just adding the exponents.', 'start': 830.843, 'duration': 6.163}, {'end': 844.07, 'text': 'Lastly, we learned that the log of division is the same as subtracting the exponents.', 'start': 838.187, 'duration': 5.883}, {'end': 847.831, 'text': "One last thing before we're all done here.", 'start': 845.91, 'duration': 1.921}, {'end': 851.673, 'text': "What we've talked about applies to all logs.", 'start': 848.872, 'duration': 2.801}, {'end': 863.127, 'text': 'This means it applies to log base 10 which is useful when things go up or down by powers of 10, like decibels or earthquakes.', 'start': 853.094, 'duration': 10.033}], 'summary': 'Symmetry in 8-fold, geometric mean for log-based data, log properties, applies to all logs including log base 10.', 'duration': 58.269, 'max_score': 804.858, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VSi0Z04fWj0/pics/VSi0Z04fWj0804858.jpg'}], 'start': 609.943, 'title': 'Logarithmic operations and understanding', 'summary': 'Explains logarithmic operations with powers of 2, demonstrating log properties for multiplication and division. it also delves into understanding logarithms, log scales, and their applications, emphasizing the benefits of using the geometric mean for log-based data.', 'chapters': [{'end': 765.772, 'start': 609.943, 'title': 'Logarithmic operations with powers of 2', 'summary': 'Explains how to use logarithms to simplify multiplication and division of numbers by rewriting them as powers of 2 and adding or subtracting their exponents, demonstrating that the log of multiplied numbers is the sum of their exponents and how division becomes subtraction through exponent subtraction.', 'duration': 155.829, 'highlights': ['The chapter illustrates that the log of multiplied numbers is just the sum of their exponents, whether the numbers are power of 2 friendly or not.', 'It explains how division becomes subtraction by rewriting the division as multiplication and subtracting the exponents after converting the numbers to powers of 2.', 'The transcript emphasizes the process of using logarithms to simplify multiplication and division by rewriting the numbers as powers of 2 and adding or subtracting their exponents.']}, {'end': 829.122, 'start': 767.264, 'title': 'Understanding logarithms and log scales', 'summary': 'Explores the concept of logarithms, emphasizing that logs isolate exponents and highlighting the symmetrical nature of log scales for plotting fold change. additionally, it discusses the benefits of using the geometric mean for log-based data, particularly in scenarios where values are doubling every time and are less sensitive to outliers.', 'duration': 61.858, 'highlights': ['Logarithms isolate exponents, as seen in the example of log base 2 of 8 equaling 3, emphasizing the relationship between logarithms and exponents.', 'Log scales are beneficial for plotting fold change due to their symmetrical nature, where 8 fold up is the same distance from 0 as 8 fold down, providing a quantifiable example of their utility.', 'The geometric mean is less sensitive to outliers than the normal mean, making it a valuable measure for log-based data, particularly when values are doubling every time, showcasing its practical advantage.']}, {'end': 936.255, 'start': 830.843, 'title': 'Understanding logarithms and their applications', 'summary': 'Explains the properties of logarithms, including the relationship between multiplication and addition of exponents, the relationship between division and subtraction of exponents, and the universality of these properties for all types of logarithms including log base 10 and log base e.', 'duration': 105.412, 'highlights': ['The properties of logarithms, including the relationship between multiplication and addition of exponents, and the relationship between division and subtraction of exponents, are applicable to all types of logarithms, such as log base 10 and log base e.', 'The natural log, represented by log base e, is approximately 2.7 and is often the default in applications like the R programming language.', 'Logarithms can be used with different bases depending on the data, such as using log to the base 3 if the data triples at each step or using log to the base 7.5 if the data goes up by 7.5 at each step.']}], 'duration': 326.312, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VSi0Z04fWj0/pics/VSi0Z04fWj0609943.jpg', 'highlights': ['The geometric mean is less sensitive to outliers than the normal mean, making it a valuable measure for log-based data, particularly when values are doubling every time, showcasing its practical advantage.', 'Log scales are beneficial for plotting fold change due to their symmetrical nature, where 8 fold up is the same distance from 0 as 8 fold down, providing a quantifiable example of their utility.', 'The chapter illustrates that the log of multiplied numbers is just the sum of their exponents, whether the numbers are power of 2 friendly or not.', 'The properties of logarithms, including the relationship between multiplication and addition of exponents, and the relationship between division and subtraction of exponents, are applicable to all types of logarithms, such as log base 10 and log base e.']}], 'highlights': ['Using log scale ensures symmetric representation of positive and negative fold changes, where 8 fold up is equidistant from 0 as 8 fold down.', 'Demonstration of how log function isolates the exponent for numbers less than 1, such as log base 2 of 1 half equals -1, log base 2 of 1 quarter equals -2, and log base 2 of 1 over 8 equals -3, highlighting the exponential difference between these numbers.', 'The geometric mean is less sensitive to outliers than the normal mean, making it a valuable measure for log-based data, particularly when values are doubling every time, showcasing its practical advantage.', 'The log base 2 scale is explained by rewriting numbers as powers of 2 and isolating the exponent using the log function, for instance, log base 2 of 8 equals 3, log base 2 of 4 equals 2, log base 2 of 2 equals 1, and log base 2 of 1 equals 0.', 'The demonstration of calculating the geometric mean with logs and its robustness to the effects of outliers compared to the mean of normal numbers.', 'Log scales are beneficial for plotting fold change due to their symmetrical nature, where 8 fold up is the same distance from 0 as 8 fold down, providing a quantifiable example of their utility.', 'The properties of logarithms, including the relationship between multiplication and addition of exponents, and the relationship between division and subtraction of exponents, are applicable to all types of logarithms, such as log base 10 and log base e.', 'The use of LogBase 2 in qPCR is explained, emphasizing that every cycle doubles the amount of transcripts, which can lead to outliers.', 'In R programming, log base 2 of 0 is defined as negative infinity, aligning with the intuition that 1 over 2 to infinity equals the smallest imaginable number, which is 0.', 'The explanation of how log base 2 makes sense in qPCR data interpretation, with examples of differences in cycles and average calculation.', 'The chapter illustrates that the log of multiplied numbers is just the sum of their exponents, whether the numbers are power of 2 friendly or not.', 'Logs isolate exponents, demonstrated by the example of log base 2 of 8 equals the log base 2 of 2 to the third, resulting in 3.']}