An easy to use calculator to change the base of a logarithmic function is presented. Also an activity related to the change of base, using this calculator, is presented.
Given \( \log_b (x) \) , we can chose any base \( B \), such that \( B \gt 0 \) and \( B \ne 1 \) and change the given base \( b \) to \( B \) as follows
\[ \log_b (x) = \dfrac{\log_B (x) }{\log_B (b) } \]
Example 1: Change the base of \( \log_2 (x) \) to the natural base \( e \).
Solution
Substitute \( b = 2 \) and \( B = e \) in the formula of the change of base.
\( \quad \log_2 (x) = \dfrac{\log_e (x) }{\log_e (2) } \)
Simplify to
\( \qquad \qquad = \dfrac{1}{ \log_e (2)} \log_e (x) \)
Example 2: Rewrite \( \log_4 (x) \) to a base equal to \(2 \).
Solution
Substitute \( b = 4 \) and \( B = 2 \) in the formula of the change of base.
\( \quad \log_4 (x) = \dfrac{\log_2 (x) }{\log_2 (4) } \)
Simplify to
\( \qquad = \dfrac{1}{ \log_2 (4)} \log_2 (x) \)
Note that
\( \quad \log_2 (4) = \log_2 (2^2) = 2\)
and hence
\( \quad \log_4 (x) = \dfrac{1}{2} \; \log_2 (x) = 0.5 \; \log_2 (x) \)
Answer
Activity 1: Change the base of the given logarithm to the given base \( B \) manually and using the calculator and compare the two results.
a) \( \quad \log_{16} (x) \) , \( B = 4 \)
b) \( \quad \log_{10} (x) \) , \( B = e \)
c) \( \quad \log_{0.1} (x) \) , \( B = 10 \)
d) \( \quad \log_2 (x) \) , \( B = 8 \)
e) \( \quad \log_8 (x) \) , \( B = 16 \)
f) \( \quad \log_{0.01} (x) \) , \( B = 10 \)
g) \( \quad \log_{25} (x) \) , \( B = 5 \)
h) \( \quad \log_{e} (x) \) , \( B = 10 \)