3.2: Logarithm Properties | Precalculus

How it works

A logarithm gives an exponent. Product and quotient rules require positive arguments over the real numbers. Apply the rules to factors, not terms of a sum. The power rule moves an exponent outside a logarithm when the relevant real-domain conditions hold.

\log_b(MN)=\log_b M+\log_b N\quad(M,N>0;\ b>0,\ b\ne1)

Worked examples

Example 1

Evaluate $\displaystyle \log_2 32$ .

Since $\displaystyle 2^5=32$ , the logarithm is $\displaystyle 5$ .

Example 2

Expand $\displaystyle \ln(x^3y)$ assuming $\displaystyle x,y>0$ .

Use product and power rules: $\displaystyle \ln(x^3y)=3\ln x+\ln y$ .

Problems

Problem 3.2.1

Condense $\displaystyle \log_2x+\log_2(x-1)$ and state the original domain.

Show worked solution

The result is $\displaystyle \log_2(x(x-1))$ , retaining $\displaystyle x>1$ because both original arguments must be positive.

Problem 3.2.2

Evaluate $\displaystyle \log_3(1/27)$ .

Show worked solution

$\displaystyle 1/27=3^{-3}$ , so the value is $\displaystyle -3$ .

Problem 3.2.3

Is $\displaystyle \ln(x+2)=\ln x+\ln2$ a valid identity for $\displaystyle x>0$ ?

Show worked solution

No. The product rule applies to $\displaystyle 2x$ , not $\displaystyle x+2$ . For example, $\displaystyle x=1$ gives $\displaystyle \ln3$ on the left and $\displaystyle \ln2$ on the right.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 3.2.1

Problem 3.2.2

Problem 3.2.3

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