Logarithms Problem set A

Solve:

1. (a) \(\log_{10} x = -3\);
2. (b) \(\log_{10}(y-4) = -2\);
3. (c) \(\log_{10}(2t+1) = 1\).

Solve. Give the exact answer, then x to 3 significant figures.

1. (a) \(\ln x = 3\);
2. (b) \(4\ln x + 10 = 0\);
3. (c) \(\ln(4x-3) = 2\).

\(\text{Write each in terms of } p = \log_{10} a \text{ and } q = \log_{10} b:\)

1. (a) \(\log_{10}\left(\frac{a^5}{b^2}\right)\);
2. (b) \(\log_{10}\left(\sqrt[3]{a^2 b}\right)\);
3. (c) \(\log_{10}\left(\frac{1000\sqrt{a}}{b^3}\right)\).

Solve. Give exact answers.

1. (a) \(\log_{10} x + \log_{10} 8 = 2\);
2. (b) \(\log_{10}(x+4) + \log_{10} 5 = 2\).

Solve. Give exact answers.

1. (a) \(\log_{10}(x+6) - \log_{10}(x-3) = 1\);
2. (b) \(\ln(2x+1) - \ln(x-1) = 3\). Give x to 3 significant figures as well.

Solve. Give the answer in terms of natural logarithms, then to 3 significant figures.

1. (a) \(4^{x-1} = 3^{x+2}\);
2. (b) \(2^{3x+1} = 7^{x}\).

Solve:

1. (a) \(10^x = 0.04\). Give x to 3 significant figures.
2. (b) \(\log_{10}(3x+1) = 2\).
3. (c) \(5\log_{10}x - 1 = 3\). Give x to 3 significant figures.

Solve:

1. (a) \(e^x = 30\). Give x to 3 significant figures.
2. (b) \(2\ln x + 1 = 7\). Give the exact value of x, then x to 3 significant figures.
3. (c) \(5 - \ln x = 1.5\). Give x to 3 significant figures.

The pH of a solution is \(\text{pH} = -\log_{10} h\), where h is the concentration of hydrogen ions in mol/dm³.

1. (a) \(\text{Find the pH when } h = 3 \times 10^{-5}.\)
2. (b) A different solution has pH 4.2. Find h.

The height of a young tree is \(H = 2 + 1.5\ln t\) metres, where t is the age in years.

1. (a) \(\text{Find the height when } t = 10.\)
2. (b) Find the age when the height is 5 m. Give an exact answer, then 3 significant figures.

The population of town X is \(P = 12\,000 \times 1.03^t\) and the population of town Y is \(Q = 18\,000 \times 0.99^t\), where t is the number of years from now. Find when the two populations are equal. Give the exact form, then t to 3 significant figures.

The magnitude of an earthquake is \(M = \log_{10}A + 1.7\), where A is the largest ground movement in micrometres.

1. (a) Find the magnitude when \(A = 4000\).
2. (b) Find A when the magnitude is 6.2. Give your answer to 3 significant figures.

The charge of a phone battery, \(C\%\), t hours after it is plugged in is modelled by \(C = 20 + 30\ln t\), for \(1 \le t \le 8\).

1. (a) Find the charge after 3 hours.
2. (b) Find the exact time when the charge is 80%, and the time to 3 significant figures.

The depth D cm of sediment on a lake bed is modelled by \(D=a+b\ln t\), where t is the time in years since the lake was cleared. The depth is 4.0 cm when \(t=3\) and 6.5 cm when \(t=20\).

1. (a) Find b and a. Give each to 3 significant figures.
2. (b) Find the depth predicted after 50 years.
3. (c) Find when the depth is 8.0 cm.

The energy E joules released by an earthquake of magnitude M satisfies \(\log_{10}E=4.8+1.5M\).

1. (a) Find the energy released by an earthquake of magnitude 6. Give your answer to 3 significant figures.
2. (b) Find the magnitude of an earthquake that releases \(2\times10^{15}\) joules.
3. (c) How many times more energy does a magnitude 8 earthquake release than a magnitude 6 earthquake?

Solve \(\log_{10} x + \log_{10}(x-3) = 1\).

Show that the solution of \(5^{x+1} = 3^{2x-1}\) can be written as \(x = \frac{\ln 15}{\ln 1.8}\). Then find x to 3 significant figures.