The common logarithm of a number (log) is the power to which 10 must be raised to equal that number. For example, the common logarithm of 100 is 2, because 10 must be raised to the second power to equal 100. Additional examples follow.
Logarithms and Exponential Numbers
| Number | Number Expressed Exponentially | Common Logarithm |
|---|---|---|
| 1000 | 103 | 3 |
| 10 | 101 | 1 |
| 1 | 100 | 0 |
| 0.1 | 10−1 | −1 |
| 0.001 | 10−3 | −3 |
What is the common logarithm of 60? Because 60 lies between 10 and 100, which have logarithms of 1 and 2, respectively, the logarithm of 60 is 1.7782; that is,
$$60=10^{1.7782}$$
The common logarithm of a number less than 1 has a negative value. The logarithm of 0.03918 is −1.4069, or
$$0.03918=10^{−1.4069}=\frac{1}{10^{1.4069}}$$
To obtain the common logarithm of a number, use the log button on your calculator. To calculate a number from its logarithm, take the inverse log of the logarithm, or calculate 10x (where x is the logarithm of the number).
The natural logarithm of a number (ln) is the power to which e must be raised to equal the number; e is the constant 2.7182818. For example, the natural logarithm of 10 is 2.303; that is,
$$10=e^{2.303}=2.7182818^{2.303}$$
To obtain the natural logarithm of a number, use the ln button on your calculator. To calculate a number from its natural logarithm, enter the natural logarithm and take the inverse ln of the natural logarithm, or calculate ex (where x is the natural logarithm of the number).
Logarithms are exponents; thus, operations involving logarithms follow the same rules as operations involving exponents.
- The logarithm of a product of two numbers is the sum of the logarithms of the two numbers. $$log(xy)=log(x)+log(y) \\ \text{and} \\ ln(xy)=ln(x)+ln(y)$$
- The logarithm of the number resulting from the division of two numbers is the difference between the logarithms of the two numbers. $$log\left(\frac{x}{y}\right)=log(x)−log(y) \\ \text{and}\\ ln\left(\frac{x}{y}\right)=ln(x)−ln(y)$$
- The logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. $$logx^n=n\cdot log(x) \\ \text{and} \\ lnx^n=n\cdot ln(x)$$