Solutions — Y11 Methods Exp/Log Unit Summary Worksheet

Section 1 — Rational exponents

Q1. Exact evaluations:

a $8$

b $9$ ($3^{2}$)

c $8$ ($2^{3}$)

d $16$ ($4^{2}$)

Q2. Simplify:

a $\dfrac{1}{x}$ ($x^{5-2-4}=x^{-1}$)

b $4a^{5}b^{2}$ ($4a^{6}b^{2}\cdot a^{-1}$)

Section 2 — Exponential graphs

Q3. Asymptote · $y$-int · range:

a $y=2^{x}-3$: asy $y=-3$ · $y$-int $(0,-2)$ · range $y>-3$

b $y=3^{x-1}+4$: asy $y=4$ · $y$-int $\big(0,\tfrac{13}{3}\big)$ ($3^{-1}+4=\tfrac{1}{3}+4=\tfrac{13}{3}$) · range $y>4$

Section 3 — Solving exponentials

Q4. Same-base:

a $x=2$ ($3^{4}$)

b $x=\tfrac{7}{2}$ ($x+1=3x-6$)

c $x=-\tfrac{7}{2}$ ($2x+2=-5$)

d $x=-4$ ($2^{-x}=2^{4}$)

Q5. Let $u=3^{x}$: $u^{2}-4u+3=0 \Rightarrow (u-1)(u-3)=0$. Both $u>0$, so:

· $x=0$ or $x=1$ ($3^{x}=1\to x=0$; $3^{x}=3\to x=1$)

Section 4 — Logarithms & log laws

Q6. Exact evaluations:

a $4$ ($3^{4}=81$)

b $-3$ ($10^{-3}=0.001$)

c $3$ ($\log_{2}\tfrac{32}{4}=\log_{2}8$)

d $2$ ($\log_{5}25$)

Q7. Combine to a single log:

a $\log_{a}(15)$ ($\log_{a}\tfrac{6\cdot 5}{2}$)

b $\log_{a}(36)$ ($\log_{a}(9)+\log_{a}(4)$)

Section 5 — Solving with logs

Q8. Take a log of both sides:

a $x=\log_{2}(100)\approx 6.64$

b $x=\log_{5}(20)-1\approx 0.86$

Q9. Combine, exponentiate, then check the domain:

a $\log_{2}\big(x(x-2)\big)=3 \Rightarrow x^{2}-2x=8 \Rightarrow (x-4)(x+2)=0$. Need $x>2$, so reject $x=-2$. $x=4$

b $\log_{3}\big((x-1)(x+1)\big)=1 \Rightarrow x^{2}-1=3 \Rightarrow x=\pm 2$. Need $x>1$, so reject $x=-2$. $x=2$

Section 6 — Log graphs & inverses

Q10. Swap, solve for $y$, state domain:

a $y=3^{x}-7\to x=3^{y}-7 \to y=\log_{3}(x+7)$. $f^{-1}(x)=\log_{3}(x+7)$, dom $x>-7$

b $y=\log_{2}(x+4)-1\to x=\log_{2}(y+4)-1 \to y+4=2^{x+1} \to y=2^{x+1}-4$. $f^{-1}(x)=2^{x+1}-4$, dom $x\in\mathbb{R}$

Challenge

Q11. $M=20\cdot\big(\tfrac{1}{2}\big)^{t/8}$:

a Half: $M=10 \Rightarrow \big(\tfrac{1}{2}\big)^{t/8}=\tfrac{1}{2} \Rightarrow \tfrac{t}{8}=1$. Half-life $=8$ years

b $M=5 \Rightarrow \big(\tfrac{1}{2}\big)^{t/8}=\tfrac{1}{4}=\big(\tfrac{1}{2}\big)^{2} \Rightarrow \tfrac{t}{8}=2$. $t=16$ years (exact & 2 d.p.)