Online Logarithm Calculator - Compute Logs and Exponential Forms
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.
Understanding the 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 Logarithm Rules: Product and Quotient
1 - Product Rule
\[ \log_b( x \cdot y) = \log_b( x ) + \log_b( y) \]
2 - Quotient Rule
\[ \log_b \left( \dfrac{x}{y} \right) = \log_b( x ) - \log_b( y) \]
How to Use the 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
Practice Activities with the Logarithm Calculator
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 |
| \( \log_b (y) \) | | | | | | | |
| \( \color{red}{\log_b (x \cdot y)} \) | | | | | | | |
| \( \color{red}{\log_b (x) + \log_b (y)} \) | | | | | | | |
| \( \color{blue}{\log_b \left(\dfrac{x}{y}\right)} \) | | | | | | | |
| \( \color{blue}{\log_b (x) - \log_b (y)} \) | | | | | | | |
Change of Base Formula – Examples and Exercises
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 |
| \( \dfrac{\log_B(x)}{\log_B(b)} \) | | | | | |
Additional References and Resources on Logarithms
logarithms
exponential
Rules of Logarithm and Exponential - Questions with Solutions
change of base formula
Maths Calculators and Solvers.