6.4 Solving equations of the form a^x = b — Practise

Work through these on your own. Isolate the exponential before you take logarithms, and check that any answer to a logarithmic equation lies in the domain.

Both the questions and the answers below come from a set that is checked by computer every time the book is built.

Solve $2^x = 7$ , giving $x$ exactly.
Solve $3^x = 20$ , giving $x$ exactly.
Solve $5^x = 0.1$ , giving $x$ exactly.
Solve $e^x = 5$ , giving $x$ exactly.
Solve $e^{2x} = 9$ , giving $x$ exactly.
Solve $e^{2x} - 5e^x + 6 = 0$ , giving each value of $x$ exactly.
Solve $2^{2x} - 5\left(2^x\right) + 4 = 0$ , giving each value of $x$ exactly.
Solve $5^{2x-1} = 30$ , giving $x$ exactly.
Solve $3^{x+1} = 100$ , giving $x$ exactly.
Solve $2^x = 7$ , giving $x$ to three significant figures.
Solve $e^x = -1$ for real $x$ , or explain why there is no solution.
Solve $4^x = 8$ by writing both sides as powers of $2$ .

Answers and concise solutions

1. The value of $x$

Answer: Take logarithms of both sides: $x\ln 2 = \ln 7$ , so $x = \dfrac{\ln 7}{\ln 2}$ . This is about $2.807$ .

2. The value of $x$

Answer: Take logarithms: $x = \dfrac{\ln 20}{\ln 3}$ , which is about $2.727$ . Any base of logarithm works; the natural logarithm is the usual choice.

3. The value of $x$

Answer: Since $0.1 = 10^{-1}$ , taking logarithms gives $x\ln 5 = -\ln 10$ , so $x = -\dfrac{\ln 10}{\ln 5} \approx -1.431$ . The answer is negative because the number on the right is smaller than $1$ .

4. The value of $x$

Answer: The natural logarithm undoes the exponential, so $x = \ln 5 \approx 1.609$ .

5. The value of $x$

Answer: Take logarithms: $2x = \ln 9$ , so $x = \dfrac{\ln 9}{2} = \ln 3 \approx 1.099$ . Halving is the last step, after the logarithm.

6. The values of $x$

Answer: This is a quadratic in $e^x$ . Factorise: $\left(e^x - 2\right)\left(e^x - 3\right) = 0$ , so $e^x = 2$ or $e^x = 3$ . Each gives a logarithm: $x = \ln 2$ or $x = \ln 3$ .

7. The values of $x$

Answer: Write $2^{2x}$ as $\left(2^x\right)^2$ and treat $2^x$ as the unknown. Factorise: $\left(2^x - 1\right)\left(2^x - 4\right) = 0$ . So $2^x = 1$ , giving $x = 0$ , or $2^x = 4$ , giving $x = 2$ .

8. The value of $x$

Answer: Take logarithms: $(2x - 1)\ln 5 = \ln 30$ , so $2x - 1 = \dfrac{\ln 30}{\ln 5}$ and $x = 1 + \dfrac{\ln 6}{2\ln 5}$ . This is about $1.557$ .

9. The value of $x$

Answer: Take logarithms: $(x + 1)\ln 3 = \ln 100$ , so $x = \dfrac{\ln 100}{\ln 3} - 1 \approx 3.192$ . The $-1$ comes from the exponent $x + 1$ .

10. The value of $x$

Answer: $x = \dfrac{\ln 7}{\ln 2} = 2.80735\ldots$ , which is $2.81$ to three significant figures.

11. The solution set

Answer: The graph of $y = e^x$ lies above the $x$ -axis for every real $x$ , so it never equals $-1$ . The solution set is empty. This is why a logarithmic equation is checked: only a positive input is allowed.

12. The value of $x$

Answer: Write $4 = 2^2$ and $8 = 2^3$ . Then $\left(2^2\right)^x = 2^3$ , so $2^{2x} = 2^3$ and $2x = 3$ . The solution is $x = \dfrac{3}{2} = 1.5$ . The computer also reports this as $\dfrac{\ln 2}{\ln 4} + 1$ , which is the same number.