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) \(\ln(4x-3) = 2\).

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 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 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?