Exp/Log Unit Summary — mixed practice
Year 11 Mathematical Methods Unit 1 · Cambridge Ch 13 (full unit) · Mr Wong
Mixed practice covering the whole unit.
- §13B rational exponents · §13C exponential graphs · §13D solving exponentials
- §13E logarithms & log laws · §13F solving with logs · §13G log graphs & inverses
- Give exact answers unless told to round. Always check the domain on log equations.
Quick reference
$a^{m/n}=\big(\sqrt[n]{a}\big)^{m}$ · $a^{-n}=\dfrac{1}{a^{n}}$ · $\log_{a}(x)=y\iff a^{y}=x$
$\log_{a}(mn)=\log_{a}m+\log_{a}n$ · $\log_{a}(m^{p})=p\log_{a}m$ · $\log_{a}(x)=\dfrac{\log_{b}x}{\log_{b}a}$
Section 1 — Rational exponents (§13B)
1. Evaluate exactly without a calculator.
a $64^{1/2}=$
b $27^{2/3}=$
c $16^{3/4}=$
d $\big(8^{2/3}\big)^{2}=$
2. Simplify using the index laws (positive indices, no fractional powers).
a $\dfrac{x^{5}\cdot x^{-2}}{x^{4}}=$
b $\big(2a^{3}b\big)^{2}\cdot a^{-1}=$
Section 2 — Exponential graphs (§13C)
3. For each, state the asymptote, the $y$-intercept and the range.
a $y=2^{x}-3$ asy: $y$-int: range:
b $y=3^{x-1}+4$ asy: $y$-int: range:
Section 3 — Solving exponentials (§13D)
4. Solve each by writing both sides with the same base.
a $3^{x+2}=81$ $x=$
b $2^{x+1}=8^{x-2}$ $x=$
c $4^{x+1}=\dfrac{1}{32}$ $x=$
d $\big(\tfrac{1}{2}\big)^{x}=16$ $x=$
5. Solve by letting $u=3^{x}$ (quadratic-in-disguise).
$9^{x}-4\cdot 3^{x}+3=0 \;\Rightarrow\; x=$ or $x=$
Exp/Log Unit Summary — continued
Logarithms · Solving with logs · Log graphs & inverses · Challenge
Section 4 — Logarithms & log laws (§13E)
6. Evaluate exactly without a calculator.
a $\log_{3}(81)=$
b $\log_{10}(0.001)=$
c $\log_{2}(32)-\log_{2}(4)=$
d $\log_{5}(50)-\log_{5}(2)=$
7. Use the log laws to write each as a single log.
a $\log_{a}(6)+\log_{a}(5)-\log_{a}(2)$
b $2\log_{a}(3)+\tfrac{1}{2}\log_{a}(16)$
Section 5 — Solving with logs (§13F)
8. Solve, giving exact answers and decimals to 2 d.p.
a $2^{x}=100$ $x=$
b $5^{x+1}=20$ $x=$
9. Solve, remembering to check the domain.
a $\log_{2}(x)+\log_{2}(x-2)=3$ $x=$
b $\log_{3}(x-1)+\log_{3}(x+1)=1$ $x=$
Section 6 — Log graphs & inverses (§13G)
10. Find the inverse function and state its domain.
a $f(x)=3^{x}-7$. $f^{-1}(x)=$ dom:
b $f(x)=\log_{2}(x+4)-1$. $f^{-1}(x)=$ dom:
Challenge
11. A radioactive sample decays so that the mass $M$ kg remaining after $t$ years satisfies $M=20\cdot \big(\tfrac{1}{2}\big)^{t/8}$. (a) Find the half-life. (b) Find $t$ when $M=5$ kg, exact and to 2 d.p.