# 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.

Answer

## 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 |

\( \dfrac{\log_B(x)}{\log_B(b)} \) |

## More References and Links

logarithmsexponential

Rules of Logarithm and Exponential - Questions with Solutions

change of base formula

Maths Calculators and Solvers .