4.2.2 Change of Base of Logarithms

\log_{a} b = \frac{\log_{c} b}{\log_{c} a} and \log_{a} b = \frac{1}{\log_{b} a}

Example:
Find the value of the following:
a.

\log_{25} 100

\log_{3} 0.45

Answer:

\begin{array}{l} (a) \log_{25} 100 = \frac{\log_{10} 100}{\log_{10} 25} = \frac{\log_{10} 10^{2}}{\log_{10} 25} = \frac{2}{1.3979} = 1.431 \\ (b) \log_{3} 0.45 = \frac{\log_{10} 0.45}{\log_{10} 3} = \frac{- 0.3468}{0.4771} = - 0.727 \end{array}

Example 1
Find the value of the following.
(a)

\log_{4} 8

(b)

\log_{125} 5

(c)

\log_{81} 27

(d)

\log_{16} 64

Example 2
Given that

\log_{p} 3 = h

and

\log_{p} 5 = k

, express the following in term of h and or k.
(a)

\log_{p} \frac{5}{3}

(b)

\log_{15} 75