Exponential & Logarithmic

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Skill: exponential-logarithmic

Exponential & Logarithmic

Exponents run the show when growth multiplies; logarithms are their undo button. Learn to evaluate powers, solve b^x = v by reading the exponent, flip between exponential and log form, and wield the three log laws to expand and condense.

1 Understand

The core idea in plain language.

What it is. b^x = v asks “which power of b gives v?” — the answer is log_b(v). A logarithm is an exponent: log_b(x) = e means b^e = x. The log laws mirror the exponent laws: log(MN) = log M + log N (product), log(M/N) = log M − log N (quotient), log(M^k) = k·log M (power).

Why it matters. Anything that multiplies per step — populations, investments, half-life — is exponential, and solving it means reading exponents with logs.

Where it is used. Compound interest · radioactive decay and carbon dating · the Richter and decibel scales (both logarithmic).

2 See It

Diagrams that make the idea visual.

Logarithm = the missing exponent

2^x = 32 asks “2 to which power is 32?” — the answer, 5, is exactly log₂(32). The log is just the question form of the exponential equation.

2? = 32 log₂(32) = 5 the log ASKS the question the power ANSWERS
2^x = 32 → x = log₂(32) = 5: the log is the missing exponent.

The three log laws

Logs turn multiplication into addition, division into subtraction, and powers into multiplication. Every expand/condense problem is one of these three moves.

log(MN) = log M + log N log(M/N) = log M − log N log(Mk) = k · log M same base throughout · M, N > 0
Product, quotient, power: the three log laws.

3 Worked Examples

Follow each step. The pattern is always the same.

Example 1 Solve: 3^x = 81
  1. Ask: 3 to which power is 81?
  2. 81 = 3⁴, so x = 4.
x = 4

Check: 3⁴ = 81 confirms it.

Example 2 Evaluate: log₅(125)
  1. Ask: 5 to which power is 125?
  2. 125 = 5³, so log₅(125) = 3.
3

Check: 5³ = 125 confirms it.

Example 3 Solve: log₂(x) = 6
  1. Flip to exponential form: x = 2⁶.
  2. x = 64 (and 64 > 0, so the domain is fine).
x = 64

Check: log₂(64) = 6 since 2⁶ = 64.

Example 4 Condense, then evaluate: 2·log₃(x) + 3·log₃(y) at x = 3, y = 1
  1. Power law: 2·log₃(x) = log₃(x²); 3·log₃(y) = log₃(y³).
  2. Product law: log₃(x²) + log₃(y³) = log₃(x²y³).
  3. At x = 3, y = 1: log₃(9·1) = log₃(9) = 2.
2

Check: 3² = 9 confirms log₃(9) = 2.

4 Common Mistakes

These errors show up on almost every quiz. Spot them now and they will never cost you points.

Mistake 1: log of a sum
Wrong
log(x + y) = log x + log y.
Right
The product law needs a PRODUCT inside: log(xy) = log x + log y. log(x + y) does not split.
Rule: logs split products and quotients — never sums.
Mistake 2: log of a negative
Wrong
log₂(−8) = −3 because 2^(−3) = −8.
Right
Undefined — 2^(−3) = 1/8, not −8. A positive base never yields a negative.
Rule: log_b(x) requires x > 0 (and b > 0, b ≠ 1).
Mistake 3: dropping the exponent
Wrong
Solving 2^x = 16 as x = 16/2 = 8.
Right
x = 4 — the exponent asks “2 to which power”, not division.
Rule: b^x = v is solved by reading the exponent (log), never by dividing.

5 Quick Check

Try each one on paper first, then reveal the answer.

1. Solve: 2^x = 64.
Answer
x = 6 since 2⁶ = 64.
2. Evaluate: log₁₀(1000).
Answer
3 — 10³ = 1000.
3. Solve: log₄(x) = 3.
Answer
x = 4³ = 64.

Key Points to Remember

  • log_b(x) = e means exactly b^e = x — the log IS the exponent.
  • b^x = v is solved by asking “b to which power is v?”
  • log(MN) = log M + log N; log(M/N) = log M − log N; log(M^k) = k·log M.
  • Logs never split sums: log(x + y) stays whole.
  • Domain: log_b(x) needs x > 0 (and b > 0, b ≠ 1).
  • Negative exponents: b^(−n) = 1/b^n.
  • Expand = laws left-to-right; condense = laws right-to-left.
Practice problems Try the tool See it on FiboMap
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