# 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.

 Base: $\quad b =$ 2 Base: $\quad B =$ 16 Decimal Places Desired = 5

## 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

logarithms
change of base formula
Rules of Logarithm and Exponential - Questions with Solutions
Maths Calculators and Solvers .