Solving Exponential Equations

Show your work for each problem.

1.
Solve $2^x = 16$ .
1. a) $4$
2. b) $8$
3. c) $14$
4. d) $2$
Answer: $4$
$16 = 2 \cdot 2 \cdot 2 \cdot 2 = 2^4$ , so $2^x = 2^4$ and $x = 4$ . The answer $8$ comes from dividing $16$ by $2$ , which is wrong.
2.
Solve $3^x = 81$ .
Answer: 4
$81 = 3^4$ , so $3^x = 3^4$ and $x = 4$ . Check: $3^4 = 81$ ✓
3.
Solve $5^{x-1} = 125$ .
Answer: 4
1. Write 125 as a power of 5: $125 = 5^{\square}$ . What is the exponent? 3
2. Set the exponents equal: $x - 1 = 3$ . What is $x$ ? 4
4.
Which equation is $4^x = 64$ written with base $2$ on both sides?
1. a) $2^{2x} = 2^6$
2. b) $2^{x+2} = 2^6$
3. c) $2^{2x} = 2^8$
4. d) $2^x = 2^6$
Answer: $2^{2x} = 2^6$
$4^x = (2^2)^x = 2^{2x}$ and $64 = 2^6$ , so the equation is $2^{2x} = 2^6$ (which gives $x = 3$ , and indeed $4^3 = 64$ ).
5.
Solve $5^x = \frac{1}{25}$ .
Answer: -2
$\frac{1}{25} = \frac{1}{5^2} = 5^{-2}$ , so $5^x = 5^{-2}$ and $x = -2$ .
6.
Solve $3 \cdot 2^x = 96$ .
Answer: 5
1. Divide both sides by 3: $2^x = \square$ 32
2. Write 32 as a power of 2. What is $x$ ? 5
7.
Solve $8^x = 4$ . Write the answer as a fraction, like 3/4.
Answer: 2/3
$8^x = (2^3)^x = 2^{3x}$ and $4 = 2^2$ , so $3x = 2$ and $x = \frac{2}{3}$ . Check: $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4$ ✓
8.
Solve $9^x = \frac{1}{27}$ .
1. a) $-3$
2. b) $-\frac{3}{2}$
3. c) $\frac{3}{2}$
4. d) $-\frac{2}{3}$
Answer: $-\frac{3}{2}$
$9^x = 3^{2x}$ and $\frac{1}{27} = 3^{-3}$ , so $2x = -3$ and $x = -\frac{3}{2}$ . The choice $\frac{3}{2}$ forgets the minus sign.
9.
Solve $2^{3x} = \frac{1}{16}$ . Write the answer as a fraction, like -5/2.
Answer: -4/3
1. Write $\frac{1}{16}$ as a power of 2: $\frac{1}{16} = 2^{\square}$ . What is the exponent? -4
2. Set the exponents equal: $3x = -4$ . What is $x$ ? Write it as a fraction. -4/3
10.
Solve $8^x = 32$ . Write the answer as a fraction, like 7/2.
Answer: 5/3
1. $8 = 2^{\square}$ . What is the exponent? 3
2. $32 = 2^{\square}$ . What is the exponent? 5
3. Now $2^{3x} = 2^5$ , so $3x = 5$ . What is $x$ ? Write it as a fraction. 5/3
11.
Solve $25^x = 125$ . Write the answer as a fraction, like 5/4.
Answer: 3/2
$25^x = (5^2)^x = 5^{2x}$ and $125 = 5^3$ , so $2x = 3$ and $x = \frac{3}{2}$ . Check: $25^{\frac{3}{2}} = (\sqrt{25})^3 = 5^3 = 125$ ✓
12.
Solve $5 \cdot 3^{x+1} = 405$ .
Answer: 3
1. Divide both sides by 5: $3^{x+1} = \square$ 81
2. $81 = 3^{\square}$ . What is the exponent? 4
3. $x + 1 = 4$ . What is $x$ ? 3
13.
The solution of $2^x = 20$ is not a whole number. Between which two consecutive integers does it lie?
1. a)4 and 5
2. b)3 and 4
3. c)10 and 11
4. d)5 and 6
Answer: 4 and 5
$2^4 = 16$ and $2^5 = 32$ . Since $16 < 20 < 32$ , the solution lies between $4$ and $5$ (it is about $4.32$ ). The choice 10 and 11 comes from wrongly dividing 20 by 2.
14.
Solve $4^{x+1} = 8^x$ .
Answer: 2
1. $4 = 2^{\square}$ . What is the exponent? 2
2. $8 = 2^{\square}$ . What is the exponent? 3
3. Now $2^{2(x+1)} = 2^{3x}$ , so $2x + 2 = 3x$ . What is $x$ ? 2
15.
The equation $2^{x^2 - x} = 64$ has two solutions. What is the larger one?
Answer: 3
1. $64 = 2^{\square}$ . What is the exponent? 6
2. So $x^2 - x - 6 = 0$ , which factors as $(x - 3)(x + 2) = 0$ . What is the smaller solution? -2
3. What is the larger solution? 3
16.
Solve $\left(\frac{1}{2}\right)^x = 8$ .
Answer: -3
$2^{-x} = 2^3$ , so $-x = 3$ and $x = -3$ . Check: $\left(\frac{1}{2}\right)^{-3} = 2^3 = 8$ ✓
17.
Which equation has no solution?
1. a) $3^x = 1$
2. b) $2^x = \frac{1}{8}$
3. c) $4^x = 2$
4. d) $5^x = -25$
Answer: $5^x = -25$
$5^x$ is positive for every $x$ , so it can never equal $-25$ . The others have solutions: $2^x = \frac{1}{8}$ gives $x = -3$ , $3^x = 1$ gives $x = 0$ and $4^x = 2$ gives $x = \frac{1}{2}$ .
18.
The solution of $3^x = 50$ is not a whole number. What is the smallest whole number $n$ with $3^n > 50$ ?
Answer: 4
1. What is $3^3$ ? 27
2. What is $3^4$ ? 81
3. Since $27 < 50 < 81$ , what is the smallest whole number $n$ with $3^n > 50$ ? 4