Logarithm Calculator

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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: \( \quad x = \)
Base: \( \quad b =\)
Decimal Places Desired =

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

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