Free preview

4.4 Laws of Logarithms

Focus on $\log$ product law and quotient law.

Before you solve

Ideas in this lesson

Explore the lesson brief and the worked example for Idea $1$ . Create a free account to continue with the remaining ideas and practice.

01Prove log laws and diagnose false identitiesWorked steps · five questions 02Evaluate combined and iterated logarithms exactlyWorked steps · five questions 03Expand logarithms without losing the real domainWorked steps · five questions 04Condense logarithms and retain excluded inputsWorked steps · five questions 05Change bases, graph logarithms and prove related identitiesWorked steps · five questions 06Recover log-linear models and interpret logarithmic differencesWorked steps · five questions 07Prove equivalent horizontal and vertical transformationsWorked steps · five questions

Lesson brief and rules

4.4 Laws of Logarithms

For positive inputs, $\log _b(MN)=\log _b M+\log _b N$ .

Worked example

1

01 / For

b>0,\ b\ne1,\ A>0,\ B>0

, prove the product, quotient and real-power laws from the exponential definition, rather than assuming them. Then explain why

\log_b(A+B)

has no corresponding sum law. Written work is required; selecting an answer alone is not a complete response. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

02 / Set

u=\log_b A,\ v=\log_b B

, so

A=b^{u},\ B=b^{v}

03 / The exponential law

A B=b^{u+v}

gives

\log_b(A B)=u+v=\log_b A+\log_b B

04 / Similarly

\dfrac{A}{B}=b^{u-v}

gives

\log_b\dfrac{A}{B}=\log_b A-\log_b B

05 / For real

r

, positivity of

A

makes

A^{r}=(b^{u})^{r}=b^{r u}

well-defined, hence

\log_b(A^{r})=r\log_b A

06 / A trial for a counterexample to an additive law is

b=2,\ A=B=2

\log_2(2+2)=2

but

\log_2 2+\log_2 2=2

happens to agree, so it is not a counterexample. Instead use

A=B=4

\log_2 8=3\ne4=\log_2 4+\log_2 4

Worked example 2 · Prove log laws and diagnose false identities

A student claims $\log_2(A-B)=\log_2A-\log_2B$ . Give a counterexample with every logarithm defined, then state the correct subtraction-of-logs law. Written work is required; selecting an answer alone is not a complete response. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

Choose $A=8,\ B=2$ . All arguments are positive.
The left side is $\log_{2}6$ and the right side is $3-1=2$ .
Since $6\ne4$ , these are unequal.
The correct identity is $\log_b A-\log_b B=\log_b\dfrac{A}{B}$ for positive arguments and a valid base.
Written-work review:

Accept mathematically equivalent derivations and valid alternative counterexamples. Every requested conclusion, original-domain restriction, justification, precision and unit requirement still applies; the listed solution is one acceptable route.
Required written evidence: Choose $A=8,\ B=2$ . All arguments are positive.
Required written evidence: The left side is $\log_{2}6$ and the right side is $3-1=2$ .
Required written evidence: Since $6\ne4$ , these are unequal.
Required written evidence: The correct identity is $\log_b A-\log_b B=\log_b\dfrac{A}{B}$ for positive arguments and a valid base.

15 questions with a worked answer for each.Start practice ↗

Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

Read this lesson preview

MathLab with Mr. Ahmed

4.4 Laws of Logarithms

Ideas in this lesson

4.4 Laws of Logarithms