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Logarithms

§ Algebra

Logarithms

CCSS.HSF.BFCCSS.HSF.LE3 min read

A logarithm is the inverse operation of exponentiation, answering the question of what power a base must be raised to in order to produce a given result. The notation log_b(x) = n means that b^n = x, where b is the base, x is the argument, and n is the result. For example, log_2(8) = 3 because 2^3 = 8.

§ 01

Why it matters

Logarithms appear throughout science, engineering, and finance where exponential relationships dominate. The Richter scale uses base-10 logarithms to measure earthquake intensity, where each whole number represents a 10-fold increase in power. Sound intensity in decibels follows a logarithmic scale, making a 60-decibel conversation 1,000 times louder than a 30-decibel whisper. In finance, compound interest calculations often require logarithms to determine how long money takes to double or triple. Computer science relies heavily on base-2 logarithms for algorithm analysis, where log_2(1024) = 10 indicates that binary search can find any item among 1,024 elements in just 10 steps. The natural logarithm appears in calculus and serves as the foundation for exponential growth models in biology and economics. Students encounter logarithmic functions in Algebra II under CCSS.HSF.BF and CCSS.HSF.LE standards.

§ 02

How to solve logarithms

Logarithms

  • log_b(x) = n means bn = x.
  • Product: log(ab) = log(a) + log(b).
  • Quotient: log(a/b) = log(a) − log(b).
  • Power: log(an) = n·log(a).

Example: log₂(8) = 3 because 2³ = 8.

§ 03

Worked examples

Beginner§ 01

log_5(125) = _______

Answer: 3

  1. Understand what a logarithm asks → log_5(125) = ? means: 5^? = 125 — A logarithm answers the question: '5 raised to WHAT power gives 125?'
  2. Try powers of 5 → 51 = 5, 52 = 25, 53 = 125 — Calculate 5^1, 5^2, ... until we reach 125.
  3. Read off the exponent → 53 = 125, so log_5(125) = 3 — The exponent that gives 125 is 3. That's our answer.
Easy§ 02

log_3(27) = _______

Answer: 3

  1. Rewrite as an exponential equation → log_3(27) = n means 3n = 27 — Converting between log form and exponential form is the key skill.
  2. Build up powers of 3 → 31 = 3, 32 = 9, 33 = 27 — Calculate successive powers of 3 until we hit 27.
  3. Identify the matching power → 33 = 27 ← match! — The 3th power of 3 equals 27.
  4. Write the answer → log_3(27) = 3 — The logarithm equals the exponent.
Medium§ 03

log_10(1002) = _______

Answer: 4

  1. Recall the power rule for logarithms → log(an) = n · log(a) — The exponent comes out as a multiplier. This is the third main log rule.
  2. Apply the rule → log_10(1002) = 2 · log_10(100) — Move the exponent 2 in front of the log.
  3. Evaluate log_10(100) → log_10(100) = 2 (since 102 = 100) — 10 raised to 2 gives 100.
  4. Multiply → 2 × 2 = 4 — Multiply the exponent by the log value.
§ 04

Common mistakes

  • Confusing the base and argument positions, writing log_8(2) = 3 instead of log_2(8) = 3 when solving 2^3 = 8.
  • Applying the product rule incorrectly by writing log_10(5 × 3) = log_10(5) × log_10(3) instead of log_10(5) + log_10(3).
  • Forgetting to apply the power rule, calculating log_2(4^3) = log_2(64) = 6 instead of using log_2(4^3) = 3 × log_2(4) = 3 × 2 = 6.
  • Mixing up logarithm properties when solving log_10(100/10) by writing log_10(100) + log_10(10) = 2 + 1 = 3 instead of log_10(100) - log_10(10) = 2 - 1 = 1.
§ 05

Frequently asked questions

What is the difference between log and ln?
The notation 'log' typically refers to the common logarithm (base 10), while 'ln' specifically means the natural logarithm (base e ≈ 2.718). Some contexts use 'log' to mean natural logarithm, so checking the base is important. Both follow the same rules and properties.
How do you check if a logarithm answer is correct?
Convert the logarithm back to exponential form and verify. If log_3(27) = 3, check that 3^3 = 27. The base raised to your answer should equal the original argument. This method works for any logarithm problem.
Why can't you take the logarithm of a negative number?
In real numbers, no positive base raised to any real power produces a negative result. Since logarithms ask 'what power gives this result?' and negative results are impossible with positive bases, logarithms of negative numbers are undefined in real number systems.
What does it mean when log_b(x) equals zero?
When log_b(x) = 0, it means b^0 = x, and since any positive number raised to the zero power equals 1, this means x = 1. Therefore, the logarithm of 1 in any base always equals zero.
How are logarithms related to exponential equations?
Logarithms are the key tool for solving exponential equations. When solving 2^x = 32, taking log_2 of both sides gives x = log_2(32) = 5. The logarithm isolates the variable from the exponent position, making exponential equations solvable.
§ 06

See also

LogarithmBaseMeanVariableExponent
§ 06

Where to next?

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