Logarithms Problem set B

Solve:

1. (a) \(\log_{10} x = -4\);
2. (b) \(\log_{10}(y-6) = -3\);
3. (c) \(\log_{10}(4t-3) = 1\).

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

1. (a) \(\ln x = -2\);
2. (b) \(6\ln x + 9 = 0\);
3. (c) \(\ln(5x+2) = 3\).

Solve:

1. (a) \(10^x = 0.7\). Give x to 3 significant figures.
2. (b) \(\log_{10}(2x-5) = 3\).
3. (c) \(4\log_{10}x + 2 = 5\). Give x to 3 significant figures.

Solve:

1. (a) \(e^x = 12\). Give x to 3 significant figures.
2. (b) \(3\ln x - 2 = 4\). Give the exact value of x, then x to 3 significant figures.
3. (c) \(8 - \ln x = 5.5\). Give x to 3 significant figures.

The loudness of a sound is \(L = 10\log_{10}\!\left(\frac{I}{10^{-12}}\right)\) decibels, where I is the intensity in W/m².

1. (a) \(\text{Find L when } I = 3 \times 10^{-5}.\)
2. (b) A different sound has loudness 65 decibels. Find I.

The depth of mud in a lake bed is \(d = 0.4 + 2.5\ln t\) cm, where t is the time in years since the lake was dug.

1. (a) \(\text{Find the depth when } t = 20.\)
2. (b) Find the time when the depth is 6 cm. Give an exact answer, then 3 significant figures.

A camera's light meter shows \(E = \log_{10}L + 2.5\), where L is the light level in lux.

1. (a) Find the meter reading when \(L = 800\).
2. (b) Find L when the reading is 7.1. Give your answer to 3 significant figures.

The depth of snow, D cm, t hours after a storm begins is modelled by \(D = 15 + 12\ln t\), for \(t \ge 1\).

1. (a) Find the depth after 5 hours.
2. (b) Find the exact time when the depth is 45 cm, and the time 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) Find h when the pH is 3.4. Give your answer to 3 significant figures.
2. (b) Find the pH when \(h=2.5\times10^{-4}\).
3. (c) How many times greater is the hydrogen ion concentration in a solution of pH 2 than in a solution of pH 5?