Logarithmic functions help us describe quantities that grow or shrink across very large ranges, such as sound intensity, earthquake strength, pH, and compound growth. A logarithm answers an exponent question: log_b(x) is the power you raise b to in order to get x. This makes logarithms the inverse of exponential functions.
Their graphs have a distinctive slow growth pattern and a vertical asymptote that shows the input must be positive.
The parent logarithmic function y = log_b(x) has domain x > 0 and passes through the point (1, 0) because b^0 = 1. For b > 1, the graph increases, but it increases more slowly as x gets larger. Common logarithms use base 10, while natural logarithms use base e, which appears often in continuous growth and decay.
Log rules let us rewrite multiplication, division, and powers in ways that make equations easier to solve.
Key Facts
- Logarithmic form and exponential form are equivalent: y = log_b(x) means b^y = x.
- For y = log_b(x), the base must satisfy b > 0 and b != 1, and the input must satisfy x > 0.
- The parent graph y = log_b(x) passes through (1, 0) because log_b(1) = 0.
- For b > 1, y = log_b(x) is increasing and has vertical asymptote x = 0.
- Common log means log(x) = log_10(x), and natural log means ln(x) = log_e(x).
- Log rules: log_b(MN) = log_b(M) + log_b(N), log_b(M/N) = log_b(M) - log_b(N), and log_b(M^p) = p log_b(M).
Vocabulary
- Logarithm
- A logarithm is the exponent needed to raise a base to a given positive number.
- Base
- The base is the number b in log_b(x) that is raised to a power.
- Natural logarithm
- The natural logarithm is a logarithm with base e, written ln(x).
- Vertical asymptote
- A vertical asymptote is a vertical line that a graph approaches but does not cross or touch.
- Inverse function
- An inverse function reverses the input and output of another function.
Common Mistakes to Avoid
- Treating log_b(x) as multiplication is wrong because log_b(x) means the exponent that produces x, not b times x.
- Taking the log of zero or a negative number is wrong in real-number algebra because logarithmic inputs must be positive.
- Forgetting the base is wrong because log_2(8), log_10(8), and ln(8) have different values.
- Using log_b(M + N) = log_b(M) + log_b(N) is wrong because log rules apply to products, quotients, and powers, not sums.
Practice Questions
- 1 Evaluate log_2(32), log_3(81), and log_10(0.01).
- 2 Solve for x: log_5(x) = 3, then solve log_4(64) = y.
- 3 Explain why the graph of y = log_b(x) has a vertical asymptote at x = 0 and why it never includes x = 0.