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) \(\ln(5x+2) = 3\).

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

1. (a) \(\log_{10}\left(\frac{a^4}{b^5}\right)\);
2. (b) \(\log_{10}\left(\sqrt[4]{a b^3}\right)\);
3. (c) \(\log_{10}\left(\frac{10\,000\sqrt[3]{b}}{a^2}\right)\).

Solve. Give exact answers.

1. (a) \(\log_{10} x + \log_{10} 4 = 3\);
2. (b) \(\log_{10}(x+1) + \log_{10} 8 = 2\).

Solve. Give exact answers.

1. (a) \(\log_{10}(x+8) - \log_{10}(x-2) = 1\);
2. (b) \(\ln(3x+2) - \ln(x-1) = 2\). Give x to 3 significant figures as well.

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

1. (a) \(5^{x-2} = 2^{x+3}\);
2. (b) \(3^{2x+1} = 11^{x}\).

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.

The number of bacteria in culture A is \(N = 500 \times 1.8^t\) and in culture B is \(M = 3200 \times 1.2^t\), where t is the time in hours. Find when the two numbers are equal. Give the exact form, then t to 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 percentage P of a hillside covered in moss is modelled by \(P=a+b\ln t\), where t is the time in years since the hillside was cleared. The cover is 12% when \(t=2\) and 30% when \(t=15\).

1. (a) Find b and a. Give each to 3 significant figures.
2. (b) Find the percentage predicted after 40 years.
3. (c) Find when the cover is 38%.

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?

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

Show that the solution of \(7^{x+1} = 2^{3x-1}\) can be written as \(x = \frac{\ln 14}{\ln(8/7)}\). Then find x to 3 significant figures.