Logarithms Problem set B One question at a time

Solve:

1. (a) \(\log_{10} x = -4\);

   • \(x = 10^{-4}\)
   • \(x = \frac{1}{10000}\)

2. (b) \(\log_{10}(y-6) = -3\);

   • \(y - 6 = 10^{-3}\)
   • \(y = \frac{6001}{1000}\)

3. (c) \(\log_{10}(4t-3) = 1\).

   • \(4t - 3 = 10^1\)
   • \(4t - 3 = 10 \Rightarrow 4t = 16\)
   • \(t = \frac{13}{4}\)

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

1. (a) \(\ln x = -2\);

   • \(x = e^{-2}\)
   • From the calculator: \(x = 0.135\)

2. (b) \(6\ln x + 9 = 0\);

   • \(6\ln x = -9 \Rightarrow \ln x = -1.5\)
   • \(x = e^{-1.5} = 0.223\)

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

   • \(5x + 2 = e^3\)
   • \(x = \frac{e^3-2}{5} = 3.62\)

Solve:

1. (a) \(10^x = 0.7\). Give x to 3 significant figures.

   • \(x = \log_{10}0.7\)
   • From the calculator: \(x = −0.155\)

2. (b) \(\log_{10}(2x-5) = 3\).

   • \(2x - 5 = 10^3 = 1000\)
   • \(2x = 1005, so x = \frac{1005}{2}\)

3. (c) \(4\log_{10}x + 2 = 5\). Give x to 3 significant figures.

   • \(4\log_{10}x = 3\)
   • \(\log_{10}x = 0.75\)
   • \(x = 10^{0.75} = 5.62\)

Solve:

1. (a) \(e^x = 12\). Give x to 3 significant figures.

   • \(x = \ln 12\)
   • From the calculator: \(x = 2.48\)

2. (b) \(3\ln x - 2 = 4\). Give the exact value of x, then x to 3 significant figures.

   • \(3\ln x = 6\)
   • \(\ln x = 2\)
   • \(x = e^2 = 7.39\)

3. (c) \(8 - \ln x = 5.5\). Give x to 3 significant figures.

   • \(\ln x = 2.5\)
   • \(x = e^{2.5} = 12.2\)

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) \(\text{Find L when } I = 3 \times 10^{-5}.\)

   • From the calculator: \(10\log_{10}(3 \times 10^{7}) = 74.8\)

2. (b) A different sound has loudness 65 decibels. Find I.

   • \(\log_{10}\!\left(\frac{I}{10^{-12}}\right) = 6.5\)
   • \(I = 10^{-12} \times 10^{6.5} = 3.16 \times 10^{-6}\)

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) \(\text{Find the depth when } t = 20.\)

   • From the calculator: \(0.4 + 2.5\ln 20 = 7.89\)

2. (b) Find the time when the depth is 6 cm. Give an exact answer, then 3 significant figures.

   • \(2.5\ln t = 5.6 \Rightarrow \ln t = 2.24\)
   • \(t = e^{2.24} = 9.39\)

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 \(L = 800\).

   • \(E = \log_{10}800 + 2.5\)
   • From the calculator: \(E = 5.40\)

2. (b) Find L when the reading is 7.1. Give your answer to 3 significant figures.

   • \(\log_{10}L + 2.5 = 7.1\), so \(\log_{10}L = 4.6\)
   • \(L = 10^{4.6} = 39 800\)

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.

   • \(D = 15 + 12\ln 5\)
   • From the calculator: \(D = 34.3\)

2. (b) Find the exact time when the depth is 45 cm, and the time to 3 significant figures.

   • \(15 + 12\ln t = 45\), so \(\ln t = 2.5\)
   • \(t = e^{2.5} = 12.2\)

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.

   • \(\log_{10}h=-3.4\)
   • \(h=10^{-3.4}=3.98\times10^{-4}\)

2. (b) Find the pH when \(h=2.5\times10^{-4}\).

   • \(\text{pH}=-\log_{10}(2.5\times10^{-4})\)
   • From the calculator: pH = 3.60

3. (c) How many times greater is the hydrogen ion concentration in a solution of pH 2 than in a solution of pH 5?

   • \(\log_{10}h_{2}-\log_{10}h_{5}=-2-(-5)=3\)
   • \(\dfrac{h_{2}}{h_{5}}=10^{3}=1000\)