The logarithmic function with base \( B \) is defined as the inverse of the exponential function with the same base.
For \( B > 0 \) and \( B \neq 1 \),
\[ y = \log_B(x) \quad \text{if and only if} \quad x = B^y \]The logarithm with base \( e \) is called the natural logarithm and is written as \( \ln(x) \).
The domain of the logarithmic function \( y = \log_B(x) \) is the set of all positive real numbers: \[ x > 0 \quad \text{or equivalently} \quad (0, +\infty) \]
Logarithms can be evaluated using their relationship with exponential functions. The logarithm represents the exponent to which the base must be raised to obtain a given number.
Since \( 2^4 = 16 \), we have \( \log_2(16) = 4 \).
Since \( \sqrt{9} = 3 \), which can be written as \( 9^{\frac{1}{2}} = 3 \), we have \( \log_9(3) = \frac{1}{2} \).
Since \( B^0 = 1 \) for \( B \neq 0 \), we have \( \log_B(1) = 0 \).
The tables and graphs below show the logarithmic functions \( \log_2(x) \), \( \log_4(x) \), and \( \log_7(x) \).
| \( x \) | \( \log_2(x) \) | \( \log_4(x) \) | \( \log_7(x) \) |
|---|---|---|---|
| \( 10^{-8} \) | \( -26.5754 \) | \( -13.2877 \) | \( -9.4663 \) |
| \( 0.001 \) | \( -9.9657 \) | \( -4.9828 \) | \( -3.5498 \) |
| \( 0.01 \) | \( -6.6438 \) | \( -3.3219 \) | \( -2.3665 \) |
| \( 0.1 \) | \( -3.3219 \) | \( -1.6609 \) | \( -1.1833 \) |
| \( 0.5 \) | \( -1 \) | \( -0.5 \) | \( -0.3562 \) |
| \( 1 \) | \( 0 \) | \( 0 \) | \( 0 \) |
| \( 2 \) | \( 1 \) | \( 0.5 \) | \( 0.3562 \) |
| \( 100 \) | \( 6.6438 \) | \( 3.3219 \) | \( 2.3665 \) |
| \( 1000 \) | \( 9.9657 \) | \( 4.9828 \) | \( 3.5498 \) |
The tables and graphs below show the logarithmic functions \( \log_{0.2}(x) \), \( \log_{0.5}(x) \), and \( \log_{0.8}(x) \).
| \( x \) | \( \log_{0.2}(x) \) | \( \log_{0.5}(x) \) | \( \log_{0.8}(x) \) |
|---|---|---|---|
| \( 10^{-8} \) | \( 11.4454 \) | \( 26.5754 \) | \( 82.5508 \) |
| \( 0.001 \) | \( 4.2920 \) | \( 9.9657 \) | \( 30.9565 \) |
| \( 0.01 \) | \( 2.8613 \) | \( 6.6438 \) | \( 20.6377 \) |
| \( 0.1 \) | \( 1.4306 \) | \( 3.3219 \) | \( 10.3188 \) |
| \( 0.5 \) | \( 0.4306 \) | \( 1 \) | \( 3.1062 \) |
| \( 1 \) | \( 0 \) | \( 0 \) | \( 0 \) |
| \( 2 \) | \( -0.4306 \) | \( -1 \) | \( -3.1062 \) |
| \( 10 \) | \( -1.4306 \) | \( -3.3219 \) | \( -10.3188 \) |
| \( 100 \) | \( -2.8613 \) | \( -6.6438 \) | \( -20.6377 \) |
From the tables and graphs above, we conclude:
To change a logarithm from base \( b \) to base \( a \), use:
\[ \log_b(x) = \frac{\log_a(x)}{\log_a(b)} \]For example: \[ \log_2(x) = \frac{\ln(x)}{\ln(2)} = \frac{\log(x)}{\log(2)} \]
In calculus, logarithms are often converted to natural logarithms because their derivatives and integrals are well studied.