Lesson 5.8
Evaluating Logs
Many logarithms can be evaluated without a calculator by recognizing powers of common bases. Mental math with logs is faster and builds deep number sense.
Introduction
If you know your powers of 2, 3, 5, and 10, you can evaluate most textbook logs instantly. The strategy: rewrite the argument as a power of the base.
Past Knowledge
Log definition (5.6), converting forms (5.7).
Today's Goal
Evaluate logs mentally by pattern recognition.
Future Success
Fast log evaluation speeds up all work in Chapter 18 (properties and solving).
Key Concepts
The Strategy
To evaluate :
Rewrite as
The exponent is the answer
Power Reference
| Base 2 | Base 3 | Base 5 | Base 10 |
|---|---|---|---|
| 2, 4, 8, 16 | 3, 9, 27, 81 | 5, 25, 125 | 10, 100, 1000 |
| 32, 64, 128 | 243, 729 | 625 | 10000 |
Worked Examples
Example 1: Powers of 2
BasicEvaluate .
Rewrite:
Example 2: Fractions → Negative Exponents
IntermediateEvaluate .
Rewrite:
Example 3: Roots → Fractional Exponents
AdvancedEvaluate .
Rewrite:
Common Pitfalls
Dividing Instead of Using Exponents
. The answer is 3, because . Don't divide — think in powers.
Missing Fractional Answers
Log answers can be fractions! because . Don't assume the answer is always an integer.
Real-Life Applications
Computer scientists constantly evaluate to determine how many binary decisions are needed. Searching a sorted list of 1,024 items requires comparisons — that's binary search.
Practice Quiz
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