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
\[ y = \log_b (x) \quad \iff \quad x = b^y \]

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.

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.