Change of Base Calculator
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.
\( \)\( \)\( \)\( \)\( \)\( \)\( \)Change of Base Formula
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
\( \quad \log_4 (x) = \dfrac{1}{2} \; \log_2 (x) = 0.5 \; \log_2 (x) \)
Use of The Change of Base Calculator
Enter the given bases \( b \) and \( B \), such that\( b \gt 0 \), \( b \ne 1 \), \( B \gt 0 \) 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
Activities Using the Change of Base Calculator
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) \( \quad \log_{e} (x) \) , \( B = 10 \)
More References and Links
logarithmschange of base formula
Rules of Logarithm and Exponential - Questions with Solutions
Maths Calculators and Solvers .
