Solutions — Exp/Log 13F Worksheet
Year 11 Methods Unit 1 · Cambridge §13F · Mr Wong
ANSWER KEY
Part A — Solve $a^x=b$
Q1. Exact $=$ fraction of logs; decimal to 2 d.p.
a $x=\dfrac{\log 5}{\log 2}\approx 2.32$
b $x=-\dfrac{\log 6}{\log 7}\approx -0.92$
c $x=\dfrac{\log 10}{\log 3}\approx 2.10$
d $x=\dfrac{\log 7}{\log 0.5}\approx -2.81$
Q2. Bracketed exponents.
a
$2x-1=\dfrac{\log 7}{\log 5}\Rightarrow$ $x\approx 1.10$
b
$1-x=\dfrac{\log 12}{\log 0.3}\approx -2.06\Rightarrow$ $x\approx 3.06$
c
$x+1=\dfrac{\log 15}{\log 2}\approx 3.91\Rightarrow$ $x\approx 2.91$
d
$2x=\dfrac{\log 20}{\log 4}\approx 2.16\Rightarrow$ $x\approx 1.08$
Part B — Inequalities
Q3. Direction flips only when dividing by $\log b$ with $0<b<1$.
a $x\le 1.66$ ($\log 4>0$, keep)
b $x>1.54$ ($\log 5>0$, keep)
c $x\le 6.64$
d $x>-1.76$ ($\log 0.4<0$, FLIP)
e $x<-3.08$ ($\log 0.7<0$, FLIP)
f $x\ge -1.89$ ($\log\tfrac{1}{3}<0$, FLIP)
Part C — Exp graphs revisited
Q4. $f(x)=2\times 10^x-4$.
· asymptote: $y=-4$
· $y$-int: $(0,-2)$
· $x$-int exact: $x=\log_{10}(2)$
· $x$-int 2 d.p.: $x\approx 0.30$
Q5. $g(x)=3\times 2^x-12$.
· asymptote: $y=-12$
· $y$-int: $(0,-9)$
· $x$-int: $3\cdot 2^x=12\Rightarrow 2^x=4\Rightarrow$ $x=2$
Challenge
Q6. $200(0.5)^{t/12}=25 \Rightarrow (0.5)^{t/12}=\tfrac{1}{8}=(0.5)^3 \Rightarrow \tfrac{t}{12}=3 \Rightarrow$ $t=36$ hours.
Q7. $200(0.5)^{t/12}=30 \Rightarrow (0.5)^{t/12}=0.15$. Take log: $\tfrac{t}{12}=\dfrac{\log 0.15}{\log 0.5}$, so $t=\dfrac{12\log 0.15}{\log 0.5}\approx 32.84$ hours.