Solutions — Exp/Log 13E Worksheet
Year 11 Methods Unit 1 · Cambridge §13E · Mr Wong
ANSWER KEY
Part A — Exp / Log form swap
Q1. Rewrite each as a log.
a $\log_5(625)=4$
b $\log_7\!\left(\tfrac{1}{49}\right)=-2$
c $\log_{1/3}\!\left(\tfrac{1}{27}\right)=3$
d $\log_{10}(1)=0$
Q2. Fill in the blanks.
a $4$ ($16=2^4$)
b $2$ ($36=6^2$)
c $-3$ ($10^{-3}$)
d arg $=125$ ($5^3$)
e arg $=\tfrac{1}{2}$ ($2^{-1}$)
f base $=9$ ($9^2=81$)
Part B — Evaluating logarithms
Q3. Rewrite the argument as a power of the base.
a $3$
b $2$
c $\tfrac{1}{2}$ ($5=25^{1/2}$)
d $\tfrac{1}{4}$ ($2=16^{1/4}$)
e $-3$
f $-4$
Q4. Fractional powers.
a $\tfrac{3}{2}$ ($8=4^{3/2}$)
b $\tfrac{4}{3}$ ($81=27^{4/3}$)
c $-\tfrac{1}{2}$
d $-\tfrac{1}{4}$
Q5. Undefined: $\log_2(-2)$ and $\log_2(0)$ (argument must be $>0$). The other two are defined.
Part C — Laws of logarithms
Q6. Single-value simplification.
a $2$ ($\log_2(4)$)
b $4$ ($4+0$)
c $\log_9\!\left(\tfrac{729}{4}\right)$ ($2-\log_9(\tfrac{4}{9})$)
d $\log_2(5x)$
Q7. Single logarithm.
a $\log_a(x^3)+\log_a(20)=$ $\log_a(20x^3)$
b $\log_2(x^2)-\log_2(x+5)=$ $\log_2\!\left(\dfrac{x^2}{x+5}\right)$
c $\log_{10}(x^9y^{18})-\log_{10}(x^5)=$ $\log_{10}(x^4y^{18})$
d $\log_7(x^2y^3)-\log_7(y^2)=$ $\log_7(x^2y)$
Q8. Change of base: $\log_2(8)=\dfrac{\log_{10}(8)}{\log_{10}(2)}=\dfrac{3\log_{10}(2)}{\log_{10}(2)}=$ $3$.
Part D — Solving log equations
Q9. Always check the domain (every log argument $>0$).
a
$\log_2(x(x+2))=3\Rightarrow x^2+2x-8=0\Rightarrow (x+4)(x-2)=0$. Reject $x=-4$. $x=2$
b
$\log_{10}\!\left(\tfrac{x+2}{x}\right)=\log_{10}(16)\Rightarrow \tfrac{x+2}{x}=16\Rightarrow 15x=2$. $x=\tfrac{2}{15}$
c
$\log_5(x(x-4))=1\Rightarrow x^2-4x-5=0\Rightarrow (x-5)(x+1)=0$. Domain $x>4$, reject $-1$. $x=5$
d
$\log_{10}(x^2)=\log_{10}(9)\Rightarrow x^2=9$. Domain $x>0$, reject $-3$. $x=3$