Logarithms are a way to describe exponents, so every logarithm asks for the power needed to make a number. They are useful because they turn multiplication into addition, division into subtraction, and powers into multiplication. These properties make complicated exponential expressions easier to solve and simplify.
Logarithms appear in science, finance, computer science, and any situation where quantities grow or shrink by repeated factors.
The main logarithm rules come directly from exponent laws. When two quantities with the same base are multiplied, their exponents add, which gives the product rule for logarithms. When a power is taken, the exponent multiplies, which gives the power rule.
The change-of-base formula lets you rewrite any logarithm using a calculator-friendly base such as 10 or e.
Key Facts
- Definition: log_b(x) = y means b^y = x, where b > 0, b != 1, and x > 0.
- Product rule: log_b(MN) = log_b(M) + log_b(N).
- Quotient rule: log_b(M/N) = log_b(M) - log_b(N).
- Power rule: log_b(M^p) = p log_b(M).
- Change of base: log_b(x) = log_a(x) / log_a(b), often log_b(x) = ln(x) / ln(b).
- Inverse facts: log_b(b^x) = x and b^(log_b(x)) = x.
Vocabulary
- Logarithm
- A logarithm is the exponent that a base must be raised to in order to produce a given positive number.
- Base
- The base is the repeated factor in an exponential expression and the number written below the log symbol in log_b(x).
- Argument
- The argument is the positive quantity inside a logarithm, such as x in log_b(x).
- Natural Logarithm
- The natural logarithm is a logarithm with base e and is written ln(x).
- Change-of-Base Formula
- The change-of-base formula rewrites a logarithm in one base as a quotient of logarithms in another base.
Common Mistakes to Avoid
- Writing log_b(M + N) = log_b(M) + log_b(N) is wrong because the product rule applies only to multiplication inside the logarithm, not addition.
- Forgetting domain restrictions is wrong because log_b(x) is defined only when the base is positive, the base is not 1, and the argument is positive.
- Changing the base incorrectly as log_b(x) = log(x) / log(x) is wrong because the denominator must be the logarithm of the original base, so log_b(x) = log(x) / log(b).
- Dropping parentheses when expanding is wrong because log_b(M/NK) may be misread; write log_b(M) - log_b(NK) or log_b(M) - log_b(N) - log_b(K) depending on the expression.
Practice Questions
- 1 Expand completely: log_3(27x^4/y^2), assuming x > 0 and y > 0.
- 2 Condense into one logarithm: 2 log_5(x) - log_5(4) + 3 log_5(y), assuming x > 0 and y > 0.
- 3 Explain why log_2(8 + 4) is not equal to log_2(8) + log_2(4), and state which logarithm rule students are often misusing.