Logarithmic and exponential equations

Unit 3 Logarithmic and exponential equations

Key unit competence: Apply exponential or logarithmic functions and equations in solving related problems.

Key ideas

• Exponential function $f(x)=a^x$ ($a>0$, $a\ne 1$): domain $\mathbb{R}$, range $]0,+\infty [$; increasing if $a>1$, decreasing if $0<a<1$. Natural base $e\approx 2.718$.
• Logarithm to base $a$: ${\log }_{a}x=y⟺a^y=x$ ($x>0$). $\ln x={\log }_{e}x$; $\log x={\log }_{10}x$.
• Rules: ${\log }_a(xy)={\log }_{a}x+{\log }_{a}y$; ${\log }_a\frac{x}{y}={\log }_{a}x-{\log }_{a}y$; ${\log }_a{x}^n=n{\log }_{a}x$; ${\log }_{a}a=1$, ${\log }_{a}1=0$; change of base ${\log }_{a}x=\frac{\ln x}{\ln a}$.
• Exponential equations: write both sides with the same base ($a^{f(x)}=a^{g(x)}⟺f(x)=g(x)$), or take logarithms of both sides. Equations like $4^x-3\cdot 2^x+2=0$: substitute $y=2^x>0$.
• Logarithmic equations: state the domain first (arguments $>0$), combine logs, convert to exponential form, then check solutions against the domain.
• Applications: growth/decay $N=N_0{e}^{kt}$, half-life, compound interest, pH $=-\log [H^{+}]$.

Formulas to know

Common misconceptions

• "$\log (x+3)=\log x+\log 3$." No: there is no rule for the log of a sum.
• "$\ln (-2)$ is negative." No: it is undefined; logarithms only take positive arguments.
• "$\frac{\log 8}{\log 2}=\log 4$." No: $\frac{\log 8}{\log 2}={\log }_{2}8=3$.

Worked examples

Practice questions

Easier and harder questions are mixed, just like in a real exam.

1. Solve $3^{x^2-1}=9^x$.

   Show answer

   $x^2-1=2x$, $x^2-2x-1=0$, $x=1\pm \sqrt{2}$.

   Common mistake: Writing $9^x=3^{x^2}$.

2. A car loses value as $V=V_0{e}^{-0.2t}$. After how many years is it worth half its price?

   Show answer

   $e^{-0.2t}=0.5$, $t=\frac{\ln 2}{0.2}\approx 3.47$ years.

   Common mistake: Sign error giving a negative time.

3. Solve ${\log }_3(2x+1)-{\log }_3(x-1)=1$.

   Show answer

   Domain $x>1$. $\frac{2x+1}{x-1}=3$, $2x+1=3x-3$, $x=4$.

   Common mistake: Subtracting the arguments instead of dividing.

4. Express ${\log }_{4}9$ in terms of ${\log }_{2}3$.

   Show answer

   ${\log }_{4}9=\frac{{\log }_{2}9}{{\log }_{2}4}=\frac{2{\log }_{2}3}{2}={\log }_{2}3$.

   Common mistake: Writing ${\log }_{4}9=\frac{9}{4}$.

5. Solve $5^{2x}-6\cdot 5^x+5=0$.

   Show answer

   $y=5^x$: $y^2-6y+5=0$, $y=1$ or 5: $x=0$ or $x=1$.

   Common mistake: Forgetting the solution $x=0$.

6. Solve $2\ln x=\ln (x+6)$.

   Show answer

   Domain $x>0$. $x^2=x+6$, $(x-3)(x+2)=0$, $x=3$.

   Common mistake: Keeping $x=-2$.

7. How long does money take to triple at 6% interest compounded continuously ($A=P{e}^{0.06t}$)?

   Show answer

   $e^{0.06t}=3$, $t=\frac{\ln 3}{0.06}\approx 18.3$ years.

   Common mistake: Using $\log 3$ with $e$.

8. The pH of a solution is 3.5. Find $[H^{+}]$.

   Show answer

   $[H^{+}]=10^{-3.5}\approx 3.16\times 10^{-4}$ mol/L.

   Common mistake: Writing $10^{3.5}$.

Matching and drag-and-drop

1. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. A substance has a half-life of 8 days. What fraction remains after 24 days?	a. $t\approx 5.36$
   2. Solve $9^x-4\times 3^x+3=0$.	b. $x=2$
   3. Solve $2^{3x-1}=32$.	c. $x=0$ or $x=1$
   4. $N=200e^{0.3t}$. When does $N$ reach 1 000?	d. $\frac{1}{8}$

   Show answer

   1 → d ($\frac{1}{8}$) · 2 → c ($x=0$ or $x=1$) · 3 → b ($x=2$) · 4 → a ($t\approx 5.36$)

   • Step 1. 24 days $=3$ half-lives.
     Step 2. ${\left(\frac{1}{2}\right)}^3=\frac{1}{8}$.
   • Step 1. $y=3^x$: $y^2-4y+3=0$, so $y=1$ or $y=3$.
     Step 2. $3^x=1$ gives 0; $3^x=3$ gives 1.
   • Step 1. $32=2^5$.
     Step 2. $3x-1=5$, so $x=2$.
   • Step 1. $e^{0.3t}=5$, so $0.3t=\ln 5$.
     Step 2. $t=\frac{1.609}{0.3}\approx 5.36$.

2. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $x=8$  ·  $x=\frac{3}{2}$  ·  $x=3$  ·  $x=16$
   1. Solve $\ln x+\ln (x-2)=\ln 3$. →
   2. Solve ${\log }_{2}x+{\log }_{4}x=6$. →
   3. Solve $\ln x-\ln (x-1)=\ln 3$. →
   Show answer

   1: $x=3$ · 2: $x=16$ · 3: $x=\frac{3}{2}$

   • Step 1. Domain $x>2$. $x(x-2)=3$.
     Step 2. $x^2-2x-3=(x-3)(x+1)=0$: only $x=3$.
   • Step 1. ${\log }_{4}x=\frac{{\log }_{2}x}{{\log }_{2}4}=\frac{1}{2}{\log }_{2}x$.
     Step 2. $\frac{3}{2}{\log }_{2}x=6$, so ${\log }_{2}x=4$ and $x=16$.
   • Step 1. Domain $x>1$. $\ln \frac{x}{x-1}=\ln 3$.
     Step 2. $x=3x-3$, so $x=\frac{3}{2}$.

3. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Find ${\log }_{5}7$ to 3 decimal places.	a. $x=4$
   2. Solve $2^x=3^{x-1}$.	b. $1.209$
   3. Solve $3\times 2^x=48$.	c. $x=7$
   4. Solve ${\log }_3(x+2)=2$.	d. $x=\frac{\ln 3}{\ln 1.5}\approx 2.71$

   Show answer

   1 → b ($1.209$) · 2 → d ($x=\frac{\ln 3}{\ln 1.5}\approx 2.71$) · 3 → a ($x=4$) · 4 → c ($x=7$)

   • Step 1. Change of base: $\frac{\ln 7}{\ln 5}$.
     Step 2. $\frac{1.9459}{1.6094}\approx 1.209$.
   • Step 1. $x\ln 2=(x-1)\ln 3$, so $x(\ln 3-\ln 2)=\ln 3$.
     Step 2. $x=\frac{\ln 3}{\ln \frac{3}{2}}\approx \frac{1.0986}{0.4055}\approx 2.71$.
   • Step 1. $2^x=16$.
     Step 2. $x=4$.
   • Step 1. $x+2=3^2=9$.
     Step 2. $x=7$ (and $x+2>0$).

4. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $x=3$  ·  $x=1$  ·  $x=\frac{\ln 2}{\ln 3-\ln 2}\approx 1.71$  ·  $x=0$
   1. Solve $e^x+e^{-x}=2$. →
   2. Solve $9^x=27^{x-1}$. →
   3. Solve $2^{x+1}=3^x$. →
   Show answer

   1: $x=0$ · 2: $x=3$ · 3: $x=\frac{\ln 2}{\ln 3-\ln 2}\approx 1.71$

   • Step 1. Multiply by $e^x$: $e^{2x}-2e^x+1=0$.
     Step 2. $(e^x-1{)}^2=0$, so $e^x=1$ and $x=0$.
   • Step 1. $3^{2x}=3^{3(x-1)}$.
     Step 2. $2x=3x-3$, so $x=3$.
   • Step 1. $(x+1)\ln 2=x\ln 3$.
     Step 2. $x(\ln 3-\ln 2)=\ln 2$, so $x=\frac{0.693}{0.405}\approx 1.71$.

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