On the page Definition of the Derivative, we have found the expression for the derivative of the natural logarithm function \(y = \ln x:\)

Now we consider the logarithmic function with arbitrary base and obtain a formula for its derivative.

So, let's take the logarithmic function \(y = {\log _a}x,\) where the base \(a\) is greater than zero and not equal to \(1:\) \(a \gt 0\), \(a \ne 1\). According to the definition of the derivative, we give an increment \(\Delta x \gt 0\) to the independent variable \(x\) assuming that \(x + \Delta x \gt 0\). The logarithmic function will increment, respectively, by the value of \(\Delta y\) where

Divide both sides by \(\Delta x:\)

Denote \({\frac{{\Delta x}}{x}} = {\frac{1}{n}}\). Then the last relation can be rewritten as

Using the power property for logarithms, we obtain:

Supposing that \(\Delta x \to 0\) (in this case \(n \to \infty\)), we find the limit of the ratio of the increments, i.e. the derivative of the logarithmic function:

Here we used the property of the limit of a composite function given that the logarithmic function is continuous. The limit in the square brackets converges to the famous trancendential number \(e\), which is approximately equal to \(2.718281828\ldots:\)

Consequently, the derivative of the logarithmic function has the form

By the change-of-base formula for logarithms, we have:

Thus,

If \(a = e\), we obtain the natural logarithm the derivative of which is expressed by the formula \({\left( {\ln x} \right)^\prime } = {\frac{1}{x}}.\)

We note another important special case − the derivative of the common logarithm (to base \(10\)):

where the number \(M\) is equal to \(M = {\log _{10}}e \approx 0.43429 \ldots \)

Note that we derived the formula \(\left( {{{\log }_a}x} \right)^\prime = \frac{1}{{x\ln a}}\) from first principles - using the limit definition of the derivative. As the logarithmic function with base \(a\) \(\left({a \gt 0}\right.\), \(\left.{a \ne 1}\right)\) and exponential function with the same base form a pair of mutually inverse functions, the derivative of the logarithmic function can also be found using the inverse function theorem.

Suppose we are given a pair of mutually inverse functions \(y = f\left( x \right) = {\log_a}x\) and \(x = \varphi \left( y \right) = {a^y}.\) Then

In the particular case \(a = e\), the derivative is given by

In the examples below, determine the derivative of the given function.

Solved Problems

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Solution.

Differentiate using the quotient rule:

where \(x \gt 0.\)

Solution.

Using the product and difference rules, we have

Solution.

Using the product rule, the chain rule and the derivative of the natural logarithm, we have

Solution.

Solution.

By the power rule and the chain rule,

Solution.

By the chain rule,