Section 4.5

Logarithmic Properties

These three properties are your main tools for solving exponential and logarithmic equations in section 4.6 — one of the most heavily tested topics on the final.

⚡ Quick Summary

Logarithms have a superpower: they turn hard operations into easy ones. Multiplying two numbers? Logs turn it into addition. Dividing? Logs turn it into subtraction. Raising to a power? Logs turn it into multiplication. Before calculators existed, this is literally how scientists did complex arithmetic — they looked up logs in a table, added them, and looked up the answer. These three shortcuts are called the product rule, quotient rule, and power rule.

Product Rule

\log_b(MN) = \log_b(M) + \log_b(N)

Multiplication inside becomes addition outside

Quotient Rule

\log_b\!\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)

Division inside becomes subtraction outside

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

When a problem says "expand," you're breaking one big log into smaller pieces — like distributing. Start by splitting fractions (quotient rule), then split multiplications (product rule), then pull exponents out front (power rule). When it says "condense," you're doing the reverse — combining several logs into one, like factoring. Move coefficients back inside as exponents first, then combine.

The change-of-base formula is your calculator lifeline. Most calculators only have $\log$ (base 10) and $\ln$ (base $e$ ) buttons. If you need $\log_5(40)$ , just type $\ln(40) \div \ln(5)$ — same answer, any base you want.

The #1 trap: thinking $\log(x + y) = \log(x) + \log(y)$ . There is no rule for the log of a sum. The addition rule only works when the numbers are *multiplied* inside the log: $\log(x \cdot y) = \log(x) + \log(y)$ . Mixing this up is the most common mistake on exams.

Key Formulas

Product Rule

\log_b(MN) = \log_b(M) + \log_b(N)

Multiplication inside becomes addition outside

Quotient Rule

\log_b\!\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)

Division inside becomes subtraction outside

Power Rule

\log_b(M^n) = n \cdot \log_b(M)

Exponent inside becomes coefficient outside

Change of Base Formula

\log_b(M) = \frac{\ln M}{\ln b} = \frac{\log M}{\log b}

Log of 1 and Log of the Base

\log_b(1) = 0 \quad\text{and}\quad \log_b(b) = 1

Useful for simplifying terms quickly

Key Takeaways

✓Product rule: $\log_b(MN) = \log_b(M) + \log_b(N)$ — multiplication inside becomes addition outside
✓Quotient rule: $\log_b(M/N) = \log_b(M) - \log_b(N)$ — division inside becomes subtraction outside
✓Power rule: $\log_b(M^n) = n \cdot \log_b(M)$ — exponents inside become coefficients outside
✓To expand: apply quotient $\to$ product $\to$ power rules. To condense: go in reverse (power $\to$ product/quotient)

⚠️ Common Mistakes

✗Writing $\log(x + y) = \log(x) + \log(y)$ — addition inside the log has NO rule; the product rule needs multiplication inside: $\log(xy)$
✗Applying the power rule backward: $\log(x^3) = 3\log(x)$ , NOT $(\log x)^3$
✗Forgetting to move coefficients back inside as exponents when condensing: $3\log(x)$ must become $\log(x^3)$ before you can combine
✗Confusing expand vs. condense direction: expanding breaks one log into many; condensing combines many logs into one

Next: Section 4.6 Builds on: 4.3

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4.4 Graphs of Logarithmic Functions

4.6 Exponential & Logarithmic Equations