# Logarithm Calculator



An easy to use calculator to compute the logarithms to any base is presented. Activities related to the product and quotient rules and the change of base formula , using the calculator, are also included.

## Logarithmic Function

The logarithmic function $\log_b \; (x)$ is defined as the inverse of the exponential function $b^x$, hence the relationship between the exponential and lagrthmic function is written as follows $y = \log_b (x) \quad \iff \quad x = b^y$

## Basic Rules of Logarithmic Function

1 - Product Rule
$\log_b( x \cdot y) = \log_b( x ) + \log_b( y)$
2 - Quotient Rule
$\log_b \left( \dfrac{x}{y} \right) = \log_b( x ) - \log_b( y)$

## Use Logarithm Calculator

Enter the argument $x$ of the logarithmic function and its base $b$ such that
$x \gt 0$ , $b \gt 0$ and $b \ne 1$
The output of the calculator is the logarithm of $x$ to the base $b$: $y = \log_b(x)$ and also its equivalent in exponential form $x = b^{y}$
NOTE that base e is entered as the letter e.

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

## Activities Using the Logarithm Calculator

Activity 1: Product and Quotient Rules
Chose any base $b$ and use the calculator to find the values of   $\log_b (x)$, $\log_b (y)$, $\log_b (x \cdot y)$, $\log_b (x) + \log_b (y)$, $\log_b \left(\dfrac{x}{y}\right)$ and $\log_b (x) - \log_b (y)$
a) Compare the quantites   $\log_b (x \cdot y)$ and   $\log_b (x) + \log_b (y)$ for each pair of values $(x,y)$. These quantities are equal according to the product rule in 1) above.
b) Compare the quantites   $\log_b \left(\dfrac{x}{y}\right)$ and   $\log_b (x) - \log_b (y)$ for each pair of values $(x,y)$. These quantities are equal according to the quotient rule in 2) above.

 $x$ 4 5 25 40 100 120 1000 $\log_b (x)$ $y$ 2 4 5 10 25 60 100 $\log_b (y)$ $\color{red}{\log_b (x \cdot y)}$ $\color{red}{\log_b (x) + \log_b (y)}$ $\color{blue}{\log_b \left(\dfrac{x}{y}\right)}$ $\color{blue}{\log_b (x) - \log_b (y)}$

Activivty 2: Change of Base Formula
The change of base formula is given by $\log_b(x) = \dfrac{\log_B(x)}{\log_B(b)}$ where $B$ is any base
Use the calculator to calculate $\log_b (x)$ and the ratio $\dfrac{\log_B(x)}{\log_B(b)}$ and compare these two quantities which according to the change of base formula are equal.

 $x$ 4 5 25 40 100 $b$ 2 3 4 5 10 $\log_b (x)$ $B$ 4 5 10 e 20 $\dfrac{\log_B(x)}{\log_B(b)}$