6.3 Logarithms and the laws of logarithms — Practise

Work through these on your own. Decide first whether the question is asking you to evaluate a logarithm, to combine logarithms, or to solve a logarithmic equation. Check that every input you use is positive.

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

Work out $\log_2 16$ .
Work out $\log_3 \dfrac{1}{27}$ .
Work out $\log_{10} 10000$ .
Write down the values of $\ln e$ , $\ln 1$ and $\ln e^4$ .
Write $5^3 = 125$ in logarithmic form.
Solve $\log_2 x = 5$ .
Solve $\log_3 x = -1$ , stating the domain you use.
Write $\ln x + \ln y$ as a single logarithm.
Write $\ln x - \ln y$ as a single logarithm.
Write $3\ln x$ as a single logarithm.
Write $-\dfrac{1}{2}\ln x$ as a single logarithm.
Expand $\ln\left(x^2y\right)$ using the laws of logarithms.
Solve $\ln x = 3$ , giving $x$ exactly.

Answers and concise solutions

1. The value

Answer: A logarithm answers the question: what power of the base gives this number? Since $2^4 = 16$ , $\log_2 16 = 4$ .

2. The value

Answer: Write $\dfrac{1}{27}$ as a power of $3$ : it is $3^{-3}$ . So $\log_3 \dfrac{1}{27} = -3$ . A logarithm of a number smaller than $1$ is negative.

3. The value

Answer: Since $10^4 = 10000$ , $\log_{10} 10000 = 4$ . The logarithm counts the number of zeros in this case.

4. The three values

Answer: $\ln e = 1$ , because $e^1 = e$ . $\ln 1 = 0$ , because $e^0 = 1$ . And $\ln e^4 = 4$ , because logarithm and exponential undo each other.

5. The value

Answer: The index becomes the logarithm: $5^3 = 125$ is the same statement as $\log_5 125 = 3$ . The base of the power is the base of the logarithm.

6. The value of $x$

Answer: Rewrite the logarithm as a power: $x = 2^5 = 32$ . The answer is positive, as every logarithm requires.

7. The value of $x$

Answer: A logarithm needs a positive input, so the domain is $x > 0$ . Rewriting gives $x = 3^{-1} = \dfrac{1}{3}$ , which is positive and so lies in the domain.

8. The single logarithm

Answer: The product law: $\ln x + \ln y = \ln(xy)$ . Adding logarithms multiplies the inputs.

9. The single logarithm

Answer: The quotient law: $\ln x - \ln y = \ln\dfrac{x}{y}$ . Subtracting logarithms divides the inputs.

10. The single logarithm

Answer: The power law: $3\ln x = \ln x^3$ . A number multiplying a logarithm becomes a power inside it.

11. The single logarithm

Answer: The power law with $k = -\dfrac{1}{2}$ : $-\dfrac{1}{2}\ln x = \ln x^{-1/2} = \ln\dfrac{1}{\sqrt{x}}$ . The law holds for negative and fractional powers too.

12. The expanded form

Answer: The product law splits $\ln(x^2y)$ into $\ln x^2 + \ln y$ . The power law then brings the $2$ down: $2\ln x + \ln y$ .

13. The value of $x$

Answer: The natural logarithm is the inverse of $e^x$ , so undo it with the exponential: $x = e^3$ . Since $e^3 \approx 20.1$ , the answer is positive, as required.