Log Rules ln x to mean log e x that is log x to the base e Natural logarithms are commonly used throughout science and engineering For example see Applications of Derivatives of Logarithms Where does this value e come from Go to Calculating the Value of e
2 Integration The Basic Logarithmic Form by M Bourne The general power formula that we saw in Section 1 is valid for all values of n except n 1 If you need a reminder about log functions check out Log base e from before Derivative of the Logarithm Function y ln x The derivative of the logarithmic function y ln x is given by
Log Rules
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Log x to mean log 10 x This is the convention used on calculators so most math text books follow along Note however that log in computer programming generally means log base e which we learn about on the next page Examples 1 Find the logarithm of 5 623 to base 10 Write this in exponential form Answer log 3 9 2 We say this as the logarithm of 9 to the base 3 is 2 What we have effectively done is to move the exponent down on to the main line This was done historically to make multiplications and divisions easier but logarithms are still very handy in mathematics The logarithmic function is defined as f x log b x
Note In Simpson s Rule n must be EVEN See below how we obtain Simpson s Rule by finding the area under each parabola and adding the areas In log log graphs both axes have a logarithmic scale The idea here is we use semilog or log log graph axes so we can more easily see details for small values of y as well as large values of y You can see some examples of semi logarithmic graphs in this YouTube Traffic Rank graph See also air pressure and Zipf Distributions later on this page
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We can expand out the log term on the right hand side as follows 3 ln 2x 3 ln 2 ln x Now the first 2 terms on the right are constants whose derivative equals zero and the derivative of the natural log of x is 1 x Then du 1 x dx t log 2 log 1 013 53 66 So it will take only about 54 years to double the world s population if it continues to grow at the current rate When the world population is 12 billion the net number of people in the world will be increasing at the rate of about 5 per second if the growth rate is still 1 3
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ln x to mean log e x that is log x to the base e Natural logarithms are commonly used throughout science and engineering For example see Applications of Derivatives of Logarithms Where does this value e come from Go to Calculating the Value of e
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2 Integration The Basic Logarithmic Form by M Bourne The general power formula that we saw in Section 1 is valid for all values of n except n 1
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Log Rules - In log log graphs both axes have a logarithmic scale The idea here is we use semilog or log log graph axes so we can more easily see details for small values of y as well as large values of y You can see some examples of semi logarithmic graphs in this YouTube Traffic Rank graph See also air pressure and Zipf Distributions later on this page
