Using Logarithms to Solve Exponentials (Ex 13F)

Year 11 Mathematical Methods Unit 1 · Cambridge §13F · Mr Wong

Name

Class

Date

Learning intentions.

Solve $a^{f(x)}=b$ by taking $\log_{10}$ of both sides and applying the power rule
Give answers in exact form and to two decimal places
Solve exponential inequalities — flip the sign when the base is between $0$ and $1$

Key results

$a^x=b \;\Rightarrow\; x=\dfrac{\log_{10}(b)}{\log_{10}(a)}$ (also written $\log_a(b)$)

Inequality: dividing by $\log_{10}(b)$ flips the direction if $0<b<1$ (since $\log_{10}(b)<0$)

Part A — Solve $a^x=b$

1. Give the exact answer (a single fraction of logs) and then to 2 d.p.

a $2^x=5$ $x=$

b $7^{-x}=6$ $x=$

c $3^x=10$ $x=$

d $0.5^x=7$ $x=$

2. Bracketed exponents — to 2 d.p.

a $5^{2x-1}=7$ $x=$

b $0.3^{1-x}=12$ $x=$

c $2^{x+1}=15$ $x=$

d $4^{2x}=20$ $x=$

Part B — Inequalities (mind the flip)

3. Solve to 2 d.p.

a $4^x\le 10$

b $5^x > 12$

c $2^x \le 100$

d $0.4^x < 5$

e $0.7^x > 3$

f $\left(\tfrac{1}{3}\right)^x \le 8$

Logs to Solve Exponentials 13F — continued

Part C (exp graphs revisited) · Challenge

Name

Part C — Exponential graphs revisited

Use logs to find the $x$-intercept that you couldn't get in §13C.

4. For $f(x)=2\times 10^x-4$, state the asymptote and find both axis intercepts (give the $x$-intercept exactly and to 2 d.p.).

· asymptote: $y=$

· $y$-int: $(0,$ $)$

· $x$-int exact: $x=$

· $x$-int 2 d.p.: $x\approx$

5. For $g(x)=3\times 2^x-12$, state the asymptote and find both axis intercepts (exact $x$-intercept).

· asymptote: $y=$

· $y$-int: $(0,$ $)$

· $x$-int: $x=$

Challenge

6. A radioactive sample of mass $M_0=200$ g decays so that $M=M_0\,(0.5)^{t/12}$ grams, with $t$ in hours. How long until only $25$ g remains? (Hint: same-base trick first, no logs needed.)

7. A different sample obeys $M=200\,(0.5)^{t/12}$. How long until $M=30$ g? Give an exact form and an answer to 2 d.p.