Logarithms (Ex 13E)

Year 11 Mathematical Methods Unit 1 · Cambridge §13E · Mr Wong

Name

Class

Date

Learning intentions.

Translate between exponential and logarithm form
Evaluate logarithms by rewriting the argument as a power of the base
Apply the three log laws (product, quotient, power) and the change-of-base rule

Key results

$a^x=y \;\Longleftrightarrow\; \log_a(y)=x$ ($a>0,\;a\ne 1,\;y>0$)

Product: $\log_a(m)+\log_a(n)=\log_a(mn)$ | Quotient: $\log_a(m)-\log_a(n)=\log_a(m/n)$

Power: $\log_a(m^p)=p\log_a(m)$ | $\log_a(1)=0$, $\log_a(a)=1$

Part A — Exp / Log form swap

1. Rewrite each exponential statement in log form.

a $5^4=625$

b $7^{-2}=\tfrac{1}{49}$

c $\left(\tfrac{1}{3}\right)^3=\tfrac{1}{27}$

d $10^0=1$

2. Fill in the blanks.

a $\log_2(16)=$

b $\log_6(36)=$

c $\log_{10}\!\left(\tfrac{1}{1000}\right)=$

d $\log_5(\;\;\;)=3$, so arg $=$

e $\log_2(\;\;\;)=-1$, arg $=$

f $\log_{\;\;\;}(81)=2$, base $=$

Part B — Evaluating logarithms

3. Evaluate.

a $\log_2(8)=$

b $\log_3(9)=$

c $\log_{25}(5)=$

d $\log_{16}(2)=$

e $\log_2\!\left(\tfrac{1}{8}\right)=$

f $\log_3\!\left(\tfrac{1}{81}\right)=$

4. Evaluate (fractional answers).

a $\log_4(8)=$

b $\log_{27}(81)=$

c $\log_{25}\!\left(\tfrac{1}{5}\right)=$

d $\log_{81}\!\left(\tfrac{1}{3}\right)=$

5. Which of the following are undefined? Circle them. $\log_2(-2)$ · $\log_2(0)$ · $\log_5(1)$ · $\log_{10}(0.001)$

Logarithms 13E — continued

Part C (laws of logs) · Part D (solve log equations)

Name

Part C — Laws of logarithms

Use the product / quotient / power laws to simplify.

6. Simplify (write your answer as a single number where possible).

a $\log_2(32)-\log_2(8)=$

b $\log_2(16)+\log_3(1)=$

c $2\log_9(9)-\log_9\!\left(\tfrac{4}{9}\right)=$

d $\log_2(5)+\log_2(x)=$

7. Write each as a single logarithm.

a $3\log_a(x)+\log_a(20)=$

b $2\log_2(x)-\log_2(x+5)=$

c $3\log_{10}(x^3y^6)-5\log_{10}(x)=$

d $\tfrac{1}{2}\log_7(x^4y^6)-2\log_7(y)=$

8. Change of base. Express $\log_2(8)$ in terms of base-$10$ logarithms and evaluate.

Part D — Solving log equations

Combine logs, rewrite in exponential form, solve, then check the domain.

9. Solve for $x$.

a $\log_2(x)+\log_2(x+2)=3$ $x=$

b $\log_{10}(x+2)-\log_{10}(x)=2\log_{10}(4)$ $x=$

c $\log_5(x)+\log_5(x-4)=1$ $x=$

d $2\log_{10}(x)=\log_{10}(9)$ $x=$