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Definition
8.5 NATURAL LOGARITHMS DEFINITION Given x > 0, the natural logarithm of x is defined as the logarithm of x to the base e. The symbol In is used for natural logarithm; thus ln x = loge x, and y = ln x if and only if x = ey. Natural logarithms are particularly convenient for problems involving derivatives and integrals. When we write lnx instead of logcx, the rules for logarithms take the following form: (i) ln 1 = 0, ln e = 1.
(ii) ln(xy) = ln x + ln y, ln(x/y) = ln x - ln y, (iii) ln(xr) = r ln x. The rules for changing the base become
Using the above equations, the formulas for the derivative and integral of bx take the form
Recall the Power Rule for integrals,
It shows how to integrate xn for n ≠ -1. Now, at long last, we are about to determine the integral of x-1. It turns out to be the natural logarithm of x.
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Last Update: 2006-11-16