Logarithmic function

Definition

A logarithmic function is the inverse of an exponential function and is typically written as $y = \log_b(x)$, where $b$ is the base. It represents the power to which the base must be raised to obtain a given number.

5 Must Know Facts For Your Next Test

1. The most common bases for logarithms are 10 (common logarithm) and e (natural logarithm).
2. Logarithmic functions have vertical asymptotes at $x = 0$ and are undefined for $x \leq 0$.
3. $\log_b(xy) = \log_b(x) + \log_b(y)$ and $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$ are key properties of logarithms.
4. The change-of-base formula allows you to convert between different bases: $\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$. This is especially useful when using calculators that only support base-10 or natural logs.
5. The graph of a logarithmic function passes through the point $(1,0)$ because any base raised to the power of zero equals one.

Review Questions

• What is the inverse function of an exponential function?
• What is the vertical asymptote of a logarithmic function?
• How can you use the change-of-base formula to calculate $\log_2(8)$ using common or natural logarithms?

Related terms

Exponential Function:

A function in which an independent variable appears in the exponent, typically written as $y = b^x$.

Natural Logarithm: $\ln(x)$ represents a logarithm with base $e$, where $e \approx 2.71828$.

Common Logarithm: $\log_{10}(x)$ represents a logarithm with base 10.