Solutions — Exp/Log 13D Worksheet
Year 11 Methods Unit 1 · Cambridge §13D · Mr Wong
ANSWER KEY
Part A — Basic same base
Q1. Rewrite the constant on the right as a power of the same base.
a $x=5$ ($32=2^5$)
b $x=4$ ($625=5^4$)
c $x=4$ ($81=3^4$)
d $x=-3$ ($\tfrac{1}{8}=2^{-3}$)
e $x=\tfrac{1}{2}$ ($2^{2x+2}=2^3$)
f $x=-\tfrac{1}{4}$ ($2^{4x}=2^{-1}$)
Part B — Rewriting both sides
Q2. Match the bases, equate exponents, solve.
a $x=-2$ ($x=2x+2$)
b $x=4$ ($x=2x-4$)
c $x=3$ ($2x=3x-3$)
d $x=\tfrac{5}{2}$ ($2x-2=3$)
e $x=3$ ($x-1=8-2x$)
f $x=1$ ($x+2=6-3x$)
Q3. Reciprocals and surds:
a
$9^{3-x}=\dfrac{1}{27^{3x}}\Rightarrow 3^{6-2x}=3^{-9x}\Rightarrow 6-2x=-9x\Rightarrow$
$x=-\tfrac{6}{7}$
b
$25^{2x-1}=5^{-1/2}\Rightarrow 5^{4x-2}=5^{-1/2}\Rightarrow 4x-2=-\tfrac{1}{2}\Rightarrow$
$x=\tfrac{3}{8}$
Part C — Quadratic in $a^x$
Q4. Substitute $u=a^x$, factorise, reject $u\le 0$, convert back.
a
$u=2^x$: $u^2+u-20=0 \Rightarrow (u-4)(u+5)=0$. Reject $u=-5$. $2^x=4\Rightarrow$ $x=2$
b
$u=5^x$: $u^2-23u-50=0 \Rightarrow (u-25)(u+2)=0$. Reject $u=-2$. $5^x=25\Rightarrow$ $x=2$
c
$u=3^x$: $u^2-4u+3=0 \Rightarrow (u-1)(u-3)=0$. Both positive. $3^x=1$ or $3^x=3 \Rightarrow$ $x=0$ or $x=1$
d
$u=2^x$: $u^2-9u+8=0 \Rightarrow (u-1)(u-8)=0$. $2^x=1$ or $2^x=8 \Rightarrow$ $x=0$ or $x=3$
Part D — Inequalities
Q5. Match the bases; preserve direction when the base is $>1$.
a $2^{4x}>2^1\Rightarrow$ $x>\tfrac{1}{4}$
b $2x-1\le 2\Rightarrow$ $x\le \tfrac{3}{2}$
c $2^{-3x+1}<2^{-4}\Rightarrow -3x+1<-4\Rightarrow$ $x>\tfrac{5}{3}$
d $3^{-x}\ge 3^2\Rightarrow -x\ge 2\Rightarrow$ $x\le -2$
e $2^x>2^3\Rightarrow$ $x>3$
f $2^{-2x}>2^5\Rightarrow -2x>5\Rightarrow$ $x<-\tfrac{5}{2}$
Challenge
Q6. $N=200\times 2^{t/3}$, solve $N=12\,800$:
·
$200\times 2^{t/3}=12800 \Rightarrow 2^{t/3}=64=2^6 \Rightarrow \tfrac{t}{3}=6 \Rightarrow$ $t=18$ hours
Q7. CAS solve $5^x=10$:
·
Exact: $x=\dfrac{\ln 10}{\ln 5}$. To 2 d.p.: $x\approx 1.43$.