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 log_{a}(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 log_{a}(x) in terms of ln(x): log_{a}(x) = ln(x)/ln(a).
Let u = log_{a}(x). This means that a^{u} = x. Now take the natural log of both sides.
log_{a}(x) = u = ln(x)/ln(a)
D_{x}(a^{log a(x)}) = D_{x}(x)
[a^{loga(x)}(ln(a))] D_{x}(log_{ a}(x)) = 1
D_{x}(log_{a}(x)) = 1/a^{log 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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