Derivative of of Logarithm to Any Base : Log_a (x)

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The steps to find the derivative of a logarithmic function to any base are presented.

Use of the Change of Base Formula

Let \( y = \log_a x \)
Use the change of base formula to rewrite \( y = \log_a x \) using the natural logarithm \( \ln \) as
\( y = \log_a x = \dfrac{\ln x}{\ln a} \)

We now evaluate the derivative
\( \dfrac {d}{dx} \left(\log_a x\right) = \dfrac {d}{dx} \left(\dfrac{ \ln x }{\ln a}\right) \)

Noting that \( \ln a \) is a constant and \( \dfrac{d}{dx} (\ln x) = \dfrac{1}{x} \), we obtain

\[ \dfrac {d}{dx} (\log_a x) = \dfrac{1}{ x \ln a} \qquad (I) \]

Examples with Solutions

Example 1

Find the derivatives of
a) \( \quad y = \log_3 (x) \)       b) \( \quad y = \log_5 (x^2 + 2x -1) \)

Solution
a) Using the formula in \( (I) \) we obtain
\( \dfrac{dy}{dx} = \dfrac{1}{x \; \ln 3} \)

b) The function in part b) is a composite function of the form \( \quad y = \log_5 u(x) \) with \( u(x) = x^2 + 2x -1 \)
Use the chain rule of differentiation , we write
\( \dfrac{dy}{dx} = \dfrac{dy}{du} \dfrac{du}{dx} \qquad (II) \)

Using the formula in \( (I) \) above
\( \dfrac{dy}{du} = \dfrac{1}{u \; \ln 5} \)

Evaluate \( \dfrac{du}{dx} \)
\( \dfrac{du}{dx} = 2 x + 2 \)

Substitute \( \dfrac{dy}{du} \) and \( \dfrac{du}{dx} \) in \( (II) \) and write

\( \dfrac{dy}{dx} = \dfrac{1}{u \; \ln 5} \; {2x+2} = \dfrac{2x+2}{ (x^2 + 2x -1) \; \ln 5} \)


Example 2

Show that the function \( \quad y = \log_{\frac{1}{2}} (x) \) is decresing in its domain.

Solution
Find the derivative using formula \( (I) \) above
\( \dfrac{dy}{dx} = \dfrac{1}{x \; \ln { \left(\frac{1}{2} \right) }} \)
Note that
\( \ln (\frac{1}{2}) = \ln 1 - \ln 2 = 0 - \ln 2 = - \ln 2 \)
Hence
\( \dfrac{dy}{dx} = \dfrac{1}{ - x \; \ln 2} \)
The domain of the given function \( \quad y = \log_{\frac{1}{2}} (x) \) is the set of all values of \( x \) such that \( x \gt 0 \) and therefore the derivative \( \dfrac{dy}{dx} = \dfrac{1}{ - x \; \ln 2} \) is negative in the domain of the given function. Since the derivative is negative over domain of the function, the given function \( \quad y = \log_{\frac{1}{2}} (x) \) is a decreasing in its domain.



More References and Links

  1. change of base formula
  2. Chain Rule of Differentiation in Calculus
  3. Convert Logarithms and Exponentials
  4. Rules of Logarithm and Exponential - Questions with Solutions

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