Logarithms (ACT + Precalculus)

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Convert between exponential and logarithmic forms.
  • Evaluate basic logarithms and apply log rules (product, quotient, power).
  • Solve simple exponential and logarithmic equations.
  • Interpret logarithms in real-world contexts (growth, orders of magnitude, pH).

Key Ideas

A logarithm answers the question: “To what exponent do we raise the base to get this number?”

The fundamental relationship: \[ \log_b(a) = c \quad \Longleftrightarrow \quad b^c = a \]

Key properties:

  • Product rule \[ \log_b(MN) = \log_b M + \log_b N \]

  • Quotient rule \[ \log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N \]

  • Power rule \[ \log_b(M^k) = k \log_b M \]

Common bases:

  • \(\log\) = base 10
  • \(\ln\) = base \(e\)
  • \(\log_2\) appears often in computer science contexts

Common Problem Types

Converting Between Forms

Rewrite \(\log_b(a)=c\) as \(b^c=a\) and vice versa.

Example:
\(\log_3(81)=4\) because \(3^4=81\).

Evaluating Basic Logs

Use known powers.

Example:
\(\log_5(125)=3\) because \(5^3=125\).

Using Log Rules

Expand or condense expressions.

Example:
\(\log(4x)=\log 4+\log x\).

Solving Exponential Equations with Logs

If bases don’t match, take logs of both sides.

Example:
\(3^x=20\) → \(x=\log_3(20)\).

Solving Log Equations

Isolate the logarithm, convert to exponential.

Example:
\(\log_2(x)=5\) → \(x=32\).

Real-World Log Interpretation

Examples include pH, decibels, earthquake magnitudes, and intensity ratios.

Example:
A one-unit increase in pH means tenfold change in hydrogen ion concentration.

Strategies

  • Convert between log and exponential form when stuck.
  • For equations, isolate logs or exponents first.
  • Apply log rules carefully: product → add, quotient → subtract, power → pull exponent down.
  • Use change of base if needed: \[ \log_b(a) = \frac{\ln a}{\ln b} \]
  • Check domains: \(\log_b(a)\) requires \(a>0\).

Worked Examples

Example 1 — Convert Between Forms

Rewrite in exponential form: \[ \log_4(64) = 3 \] This means: \[ 4^3 = 64 \]

Example 2 — Solve an Exponential Equation

Solve: \[ 5^x = 70 \]

Take \(\log\) on both sides: \[ x \log 5 = \log 70 \]

So: \[ x = \frac{\log 70}{\log 5} \]

Example 3 — Expand Using Log Rules

Expand: \[ \log(18x^2) \]

Break it apart: \[ \log 18 + \log x^2 \]

Apply power rule: \[ \log 18 + 2\log x \]

Example 4 — Solve a Log Equation

Solve: \[ \log_3(x - 2) = 4 \]

Convert: \[ x - 2 = 3^4 = 81 \]

Final: \[ x = 83 \]

WarningCommon Mistakes
  • Forgetting that logs require positive inputs.
  • Mixing up log rules (e.g., thinking \(\log(M+N)=\log M+\log N\) — it does NOT).
  • Dropping parentheses in log equations.
  • Forgetting to isolate the log before converting to exponential form.
  • Treating \(\log_b(a)\) as \(a\log b\) (incorrect).

Practice Problems

  1. Convert to exponential form: \(\log_7(49)=2\).
  2. Evaluate: \(\log_2(1/8)\).
  3. Expand: \(\log(5x^3)\).
  4. Solve: \(4^x = 90\).
  5. Solve: \(\log_5(x+4)=2\).

1.
\(7^2=49\)

2.
\(1/8 = 2^{-3}\)
So \(\log_2(1/8) = -3\)

3.
\(\log 5 + \log x^3 = \log 5 + 3\log x\)

4.
Take logs:
\(x\log 4 = \log 90\)
\[ x = \frac{\log 90}{\log 4} \]

5.
Convert:
\(x+4 = 5^2 = 25\)
So \(x = 21\)

Summary

  • Logarithms answer “what exponent gives this number?”
  • Convert freely between log and exponential forms.
  • Use product, quotient, and power rules.
  • Solve equations by isolating logs or exponents.
  • Logs appear naturally in growth, decay, and scale models.
  • If stuck: rewrite \(\log_b(a)=c\) as \(b^c=a\).
  • Use log rules to simplify before solving.
  • Check domain: arguments of logs must be positive.
  • Use change of base when needed: \(\log_b(a)=\frac{\ln a}{\ln b}\).