If a is a positive real number other than 1, then the graph of the exponential function with base a passes the horizontal line test. In this case, the inverse of the exponential function with base a is called the logarithmic function with base a, and is denoted loga(x). In particular, the natural logarithm is the logarithmic function with base e.
There are at least two ways to verify this differentiation formula.
Approach #1. Express loga(x) in terms of ln(x): loga(x) = ln(x)/ln(a).
Let u = loga(x). This means that au = x. Now take the natural log of both sides.
loga(x) = u = ln(x)/ln(a)
Dx(alog a(x)) = Dx(x)
[aloga(x)(ln(a))] Dx(log a(x)) = 1
Dx(loga(x)) = 1/alog a(x)(ln(a)) = 1/xln(a)
Combining the derivative formula for logarithmic functions, we record the following formula for future use. Here, a is a fixed positive real number other than 1 and u is a differentiable function of x.
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William A. Bogley