Overview
This lesson explains how to evaluate logarithms by relating them to exponents and patterns, including key properties, common bases, and practice problems.
Basics of Evaluating Logarithms
- A logarithm answers: "To what power must the base be raised to get this number?"
- log base 2 of 16 = 4 because 2⁴ = 16.
- log base 3 of 27 = 3 because 3³ = 27.
- log base 5 of 25 = 2 because 5² = 25.
- log base 4 of 1 = 0 because any number to the 0 power is 1.
- log base 7 of 7 = 1 because 7¹ = 7.
- log of 1000 (no base) = 3 by default, base is 10 (10³ = 1000).
- log of 100 (no base) = 2 because 10² = 100.
- log of 0.1 = -1 because 10⁻¹ = 0.1.
- In general, log of 1 is always 0 and log base b of b is always 1.
Recognizing Patterns and Shortcuts
- log of numbers with n zeros (base 10): answer is n (e.g., log 1,000,000 = 6).
- log numbers less than 1 (base 10): answer is negative (e.g., log 0.01 = -2).
- If form is "a log base a of b," the answer is b (e.g., 7 log₇ 38 = 38).
- Reversing base and argument in logs produces a reciprocal in the answer.
Exponents and Fractional Logs
- log base 3 of 9 = 2 because 3² = 9.
- log base 3 of 1/9 = -2 because 3⁻² = 1/9.
- log base 9 of 3 = 1/2 because 9¹⁄² = 3.
- log base 9 of 1/3 = -1/2 because 9⁻¹⁄² = 1/3.
- log base 2 of 32 = 5 because 2⁵ = 32.
- log base 2 of 1/32 = -5 because 2⁻⁵ = 1/32.
- log base 32 of 2 = 1/5 because 32¹⁄⁵ = 2.
- log base 32 of 1/2 = -1/5 because 32⁻¹⁄⁵ = 1/2.
Practice Examples
- 2 to what power is 8? Answer: 3 (2³ = 8).
- 3⁻² = 1/9; so log base 3 of 1/9 = -2.
- 25¹⁄² = 5; so log base 25 of 5 = 1/2.
- 64¹⁄² = 8; so log base 64 of 8 = 1/2.
- 16¹⁄⁴ = 2; so log base 16 of 2 = 1/4.
- 81¹⁄⁴ = 3; so log base 81 of 3 = 1/4.
Key Terms & Definitions
- Logarithm (log) — The power to which a base must be raised to get a certain number.
- Base (of logarithm) — The number being raised to a power in the log expression.
- Exponent — The power to which the base is raised.
Action Items / Next Steps
- Practice evaluating logarithms, especially with negative exponents and fractional answers.
- Remember log with no base means base 10.
- Complete any related homework problems given.