4.2.2 Change of Base of Logarithms

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.

$\log_{25} 100$
b.

$\log_{3} 0.45$

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)

$\log_{15} 75$

Change of Base of Logarithms

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