Logarithms Problem set A Worked solutions

Solve:

1. (a) \(\log_{10} x = -3\);

   • \(x = 10^{-3}\)
   • \(x = \frac{1}{1000}\)

2. (b) \(\log_{10}(y-4) = -2\);

   • \(y - 4 = 10^{-2}\)
   • \(y = \frac{401}{100}\)

3. (c) \(\log_{10}(2t+1) = 1\).

   • \(2t + 1 = 10^1\)
   • \(2t + 1 = 10 \Rightarrow 2t = 9\)
   • \(t = \frac{9}{2}\)

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

1. (a) \(\ln x = 3\);

   • \(x = e^3\)
   • From the calculator: \(x = 20.1\)

2. (b) \(4\ln x + 10 = 0\);

   • \(4\ln x = -10 \Rightarrow \ln x = -2.5\)
   • \(x = e^{-2.5} = 0.0821\)

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

   • \(4x - 3 = e^2\)
   • \(x = \frac{e^2+3}{4} = 2.60\)

Solve:

1. (a) \(10^x = 0.04\). Give x to 3 significant figures.

   • \(x = \log_{10}0.04\)
   • From the calculator: \(x = −1.40\)

2. (b) \(\log_{10}(3x+1) = 2\).

   • \(3x + 1 = 10^2 = 100\)
   • \(3x = 99, so x = 33\)

3. (c) \(5\log_{10}x - 1 = 3\). Give x to 3 significant figures.

   • \(5\log_{10}x = 4\)
   • \(\log_{10}x = 0.8\)
   • \(x = 10^{0.8} = 6.31\)

Solve:

1. (a) \(e^x = 30\). Give x to 3 significant figures.

   • \(x = \ln 30\)
   • From the calculator: \(x = 3.40\)

2. (b) \(2\ln x + 1 = 7\). Give the exact value of x, then x to 3 significant figures.

   • \(2\ln x = 6\)
   • \(\ln x = 3\)
   • \(x = e^3 = 20.1\)

3. (c) \(5 - \ln x = 1.5\). Give x to 3 significant figures.

   • \(\ln x = 3.5\)
   • \(x = e^{3.5} = 33.1\)

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

1. (a) \(\text{Find the pH when } h = 3 \times 10^{-5}.\)

   • From the calculator: \(-\log_{10}(3 \times 10^{-5}) = 4.52\)

2. (b) A different solution has pH 4.2. Find h.

   • \(\log_{10} h = -4.2\)
   • \(h = 10^{-4.2} = 6.31 \times 10^{-5}\)

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

1. (a) \(\text{Find the height when } t = 10.\)

   • From the calculator: \(2 + 1.5\ln 10 = 5.45\)

2. (b) Find the age when the height is 5 m. Give an exact answer, then 3 significant figures.

   • \(1.5\ln t = 3 \Rightarrow \ln t = 2\)
   • \(t = e^2 = 7.39\)

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 \(A = 4000\).

   • \(M = \log_{10}4000 + 1.7\)
   • From the calculator: \(M = 5.30\)

2. (b) Find A when the magnitude is 6.2. Give your answer to 3 significant figures.

   • \(\log_{10}A + 1.7 = 6.2\), so \(\log_{10}A = 4.5\)
   • \(A = 10^{4.5} = 31 600\)

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.

   • \(C = 20 + 30\ln 3\)
   • From the calculator: \(C = 53.0\)

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

   • \(20 + 30\ln t = 80\), so \(\ln t = 2\)
   • \(t = e^2 = 7.39\)

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.

   • \(\log_{10}E=4.8+1.5\times6=13.8\)
   • \(E=10^{13.8}=6.31\times10^{13}\)

2. (b) Find the magnitude of an earthquake that releases \(2\times10^{15}\) joules.

   • \(\log_{10}(2\times10^{15})=15.30\)
   • \(1.5M=15.30-4.8\), so \(M=7.00\)

3. (c) How many times more energy does a magnitude 8 earthquake release than a magnitude 6 earthquake?

   • \(\log_{10}E_8-\log_{10}E_6=1.5\times2=3\)
   • \(\dfrac{E_8}{E_6}=10^{3}=1000\)