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.

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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

\( \quad \log_4 (x) = \dfrac{1}{2} \; \log_2 (x) = 0.5 \; \log_2 (x) \)

\( b \gt 0 \), \( b \ne 1 \), \( B \gt 0 \) and \( B \ne 1 \)

The output of the calculator is \( \dfrac{1}{\log_B(b) } \log_b(x) \).

NOTE The letter \( e \) may be used to enter the natural base.

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) \( \quad \log_{e} (x) \) , \( B = 10 \)

change of base formula

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