Solving Exponential Equations & Inequalities (Ex 13D)

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

Name

Class

Date

Learning intentions.

Solve $a^{f(x)}=a^{g(x)}$ by rewriting both sides with the same base and equating indices
Recognise and solve "quadratic in $a^x$" problems using a substitution $u=a^x$
Solve simple exponential inequalities, choosing the correct direction for the base

Key results

If $a>0,\;a\ne 1$: $a^x=a^y \;\Leftrightarrow\; x=y$

If $a>1$: $a^x>a^y \Leftrightarrow x>y$ | If $0<a<1$: $a^x>a^y \Leftrightarrow x<y$

Part A — Basic same base

1. Solve each equation. Show one line of "rewrite with same base" working.

a $2^x=32$ $x=$

b $5^x=625$ $x=$

c $3^x=81$ $x=$

d $2^x=\tfrac{1}{8}$ $x=$

e $4^{x+1}=8$ $x=$

f $16^x=\tfrac{1}{2}$ $x=$

Part B — Rewriting both sides

2. Solve each equation.

a $2^x=4^{x+1}$ $x=$

b $3^x=9^{x-2}$ $x=$

c $9^x=27^{x-1}$ $x=$

d $25^{x-1}=125$ $x=$

e $7\times 7^{x-2}=49^{4-x}$ $x=$

f $25\times 5^x=125^{2-x}$ $x=$

3. Reciprocals and surds.

a $9^{3-x}=\dfrac{1}{27^{3x}}$ $x=$

b $25^{2x-1}=\dfrac{1}{\sqrt{5}}$ $x=$

Exp Equations 13D — continued

Part C (quadratics in disguise) · Part D (inequalities)

Name

Part C — Quadratic in $a^x$

Use the substitution $u=a^x$. Show the quadratic step.

4. Solve.

a $4^x+2^x-20=0$ $x=$

b $25^x-23(5^x)-50=0$ $x=$

c $9^x-4(3^x)+3=0$ $x=$

d $4^x-9(2^x)+8=0$ $x=$

Part D — Inequalities

State each solution as a simple inequality in $x$.

5. Solve.

a $16^x > 2$

b $3^{2x-1}\le 9$

c $2^{-3x+1}<\tfrac{1}{16}$

d $\left(\tfrac{1}{3}\right)^x\ge 9$

e $2^x > 8$

f $\left(\tfrac{1}{4}\right)^x > 32$

Challenge

6. A bacterial colony grows so that the number of cells $N$ satisfies $N=200\times 2^{t/3}$, with $t$ in hours. Find $t$ when $N=12\,800$. (Force the same base.)

7. Use CAS to solve $5^x=10$ correct to 2 decimal places, then write the exact answer in the form $x=\dfrac{\ln 10}{\ln 5}$.