Mixed exponent and logarithm practice Problem set A One question at a time

Given that \(x=\log_{10}a\), \(y=\log_{10}b\) and \(z=\log_{10}c\), write in terms of x, y and z:

1. (a) \(\log_{10}(a^3b)\)

   • \(\log_{10}(a^3b)=\log_{10}a^3+\log_{10}b\)
   • \(=3x+y\)

2. (b) \(\log_{10}\!\left(\dfrac{b^2}{ac^4}\right)\)

   • \(\log_{10}\dfrac{b^2}{ac^4}=2\log_{10}b-\log_{10}a-4\log_{10}c\)
   • \(=2y-x-4z\)

3. (c) \(\log_{10}\!\left(\dfrac{1000\sqrt b}{a^2}\right)\)

   • \(\log_{10}1000=3\) and \(\log_{10}\sqrt b=\tfrac12y\)
   • \(\log_{10}\dfrac{1000\sqrt b}{a^2}=3+\tfrac12y-2x\)

Write each expression as a single logarithm.

1. (a) \(4\ln a+2\ln b\)

   • \(4\ln a=\ln a^4\) and \(2\ln b=\ln b^2\)
   • \(\ln a^4+\ln b^2=\ln(a^4b^2)\)

2. (b) \(\tfrac12\ln x-3\ln y\)

   • \(\tfrac12\ln x=\ln\sqrt x\) and \(3\ln y=\ln y^3\)
   • \(\ln\sqrt x-\ln y^3=\ln\dfrac{\sqrt x}{y^3}\)

3. (c) \(\ln12-\ln4+2\ln3\)

   • \(\ln12-\ln4=\ln3\) and \(2\ln3=\ln9\)
   • \(\ln3+\ln9=\ln27\)

Solve each equation.

1. (a) \(\log_{10}(x+5)=2\)

   • \(x+5=10^2=100\)
   • x = 95

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

   • \(3x-2=10^1=10\)
   • 3x = 12, so x = 4

3. (c) \(\ln(x-1)=2\)

   • \(x-1=e^2\)
   • x = 1 + e² = 8.39

Use logarithms to solve each equation. Give x to 3 significant figures.

1. (a) \(7^x=30\)

   • \(x\ln7=\ln30\)
   • x = ln 30 / ln 7 = 1.75

2. (b) \(3\times5^x+4=50\)

   • \(3\times5^x=46\), so \(5^x=\dfrac{46}{3}\)
   • \(x\ln5=\ln\dfrac{46}{3}\)
   • x = 1.70

3. (c) \(2.5\times1.07^x=40\)

   • \(1.07^x=\dfrac{40}{2.5}=16\)
   • \(x\ln1.07=\ln16\)
   • x = 41.0

Find the exact value of each logarithm.

1. (a) \(\log_{10}\dfrac{1}{\sqrt{1000}}\)

   • \(\dfrac1{\sqrt{1000}}=10^{-3/2}\)
   • \(\log_{10}10^{-3/2}=-\tfrac32\)

2. (b) \(\log_{10}\sqrt[3]{100}\)

   • \(\sqrt[3]{100}=10^{2/3}\)
   • \(\log_{10}10^{2/3}=\tfrac23\)

3. (c) \(\ln\!\left(e^{2}\sqrt{e}\right)\)

   • \(e^2\sqrt e=e^{2}\times e^{1/2}=e^{5/2}\)
   • \(\ln e^{5/2}=\tfrac52\)

Solve each equation for x, where \(x>0\) and \(x\ne1\).

1. (a) \(\log_{x}81=4\)

   • \(x^4=81\)
   • \(x=81^{1/4}=3\)

2. (b) \(\log_{x}49=-2\)

   • \(x^{-2}=49\)
   • \(x^2=\tfrac1{49}\), so \(x=\tfrac17\)

3. (c) \(\log_{x}8=\tfrac32\)

   • \(x^{3/2}=8\)
   • \(x=8^{2/3}=(2^3)^{2/3}=4\)

Solve each equation by writing both sides as powers of the same base.

1. (a) \(4^{2x-1}=32^{x+2}\)

   • \(4^{2x-1}=2^{4x-2}\) and \(32^{x+2}=2^{5x+10}\)
   • 4x − 2 = 5x + 10
   • x = −12

2. (b) \(27^{x+1}=9^{2x-3}\)

   • \(27^{x+1}=3^{3x+3}\) and \(9^{2x-3}=3^{4x-6}\)
   • 3x + 3 = 4x − 6
   • x = 9

3. (c) \(3^{2x-1}=\dfrac1{81}\)

   • \(\tfrac1{81}=3^{-4}\)
   • 2x − 1 = −4
   • \(x=-\tfrac32\)

Solve \(2^{3x-1}=5^{x+2}\), giving x in the form \(\dfrac{\ln p}{\ln q}\). Then give x to 3 significant figures.

• \((3x-1)\ln2=(x+2)\ln5\)
• \(3x\ln2-\ln2=x\ln5+2\ln5\)
• \(x(3\ln2-\ln5)=2\ln5+\ln2=\ln50\)
• \(3\ln2-\ln5=\ln\tfrac85\), so \(x=\dfrac{\ln50}{\ln1.6}\)=8.32

Solve each equation.

1. (a) \(\log_{10}(4x) = 2\)

   • \(4x = 10^2 = 100\)
   • x = 25

2. (b) \(\log_{10}(x-7) = -1\)

   • \(x - 7 = 10^{-1} = 0.1\)
   • x = 7.1

3. (c) \(\ln(3x-2) = 2.5\)

   • \(3x - 2 = e^{2.5}\)
   • \(x = \frac{e^{2.5} + 2}{3} = 4.73\)

Solve each equation by writing both sides as powers of the same base.

1. (a) \(5^{x-2} = 125\)

   • \(125 = 5^3\), so \(x - 2 = 3\)
   • x = 5

2. (b) \(9^{x} = 27\)

   • \(9 = 3^2\) and \(27 = 3^3\), so \(3^{2x} = 3^3\)
   • \(2x = 3\), so \(x = \frac{3}{2}\)

3. (c) \(2^{3x+1} = 16^{x-3}\)

   • \(16 = 2^4\), so \(16^{x-3} = 2^{4(x-3)}\)
   • \(3x + 1 = 4x - 12\)
   • x = 13

The mass of a drug in a patient's blood is modelled by \(M=80\times0.85^{t}\) mg, where t is the time in hours after the dose.

1. (a) Find the mass at the start.

   • t = 0: \(M=80\times0.85^{0}=80\)

2. (b) Find the mass after 3 hours.

   • \(M=80\times0.85^{3}\)
   • From the calculator: 49.1 mg

3. (c) Use your calculator to find when the mass has fallen to 20 mg.

   • \(80\times0.85^{t}=20\)
   • Solve for t: t = 8.53 hours

The area covered by two algae in a pond, in cm², after t days is \(A=40\times1.25^{t}\) and \(B=90\times1.1^{t}\). Find when the two areas are equal. Give your answer to 3 significant figures.

• \(40\times1.25^{t}=90\times1.1^{t}\)
• \(\left(\dfrac{1.25}{1.1}\right)^{t}=\dfrac{90}{40}=2.25\)
• \(t=\dfrac{\ln2.25}{\ln(1.25/1.1)}\)
• t = 6.34 days

The amount of data sent over a network doubles every 18 months. Find how long it takes to multiply by 50. Give your answer in years to 3 significant figures.

• After t years the amount is multiplied by \(2^{t/1.5}\)
• \(2^{t/1.5}=50\)
• \(\dfrac{t}{1.5}=\dfrac{\ln50}{\ln2}\)
• t = 8.47 years

The number of bacteria in a dish is modelled by \(B=250\times1.6^{t}\), where t is the time in hours.

1. (a) Write down the number of bacteria at the start, and find the number after 5 hours. Give your answer to 3 significant figures.

   • t = 0: \(B=250\times1.6^{0}=250\)
   • \(B=250\times1.6^{5}\)
   • From the calculator: 2620

2. (b) Use your calculator to find when there are 10 000 bacteria. Give t to 3 significant figures.

   • \(250\times1.6^{t}=10\,000\)
   • Solve for t: t = 7.85 hours

3. (c) Write down the percentage increase each hour.

   • Each hour the number is multiplied by 1.6.
   • 1.6 means an increase of 60%.

The amount of carbon-14 in a bone is modelled by \(C=C_0e^{-0.000121t}\), where t is the time in years since the animal died.

1. (a) Find the half-life of carbon-14. Give your answer to 3 significant figures.

   • \(e^{-0.000121t}=0.5\)
   • \(t=\dfrac{\ln2}{0.000121}=5730\) years

2. (b) A bone has 30% of the carbon-14 that a living bone has. Find its age to 3 significant figures.

   • \(e^{-0.000121t}=0.30\)
   • \(t=\dfrac{\ln(1/0.30)}{0.000121}=9950\) years

The number of fish in a lake is modelled by \(P=P_0e^{kt}\), where t is the time in years. There are 1200 fish at \(t=0\) and 1560 fish at \(t=3\).

1. (a) Find k to 3 significant figures.

   • \(1200e^{3k}=1560\), so \(e^{3k}=1.3\)
   • \(k=\dfrac{\ln1.3}{3}=0.0875\)

2. (b) Find when there are 5000 fish. Give your answer to 3 significant figures.

   • \(1200e^{kt}=5000\), so \(e^{kt}=\dfrac{5000}{1200}\)
   • \(t=\dfrac{\ln(25/6)}{k}=16.3\) years

Given that \(4\times10^{x}=30\times5^{x}\):

1. (a) write down the exact value of \(2^x\);

   • \(10^x=2^x\times5^x\)
   • \(4\times2^x\times5^x=30\times5^x\)
   • \(2^x=\dfrac{30}{4}=\dfrac{15}2\)

2. (b) hence find x.

   • \(x=\dfrac{\ln7.5}{\ln2}\)
   • x = 2.91

Solve each pair of simultaneous equations.

1. (a) \(\log_{10}x+\log_{10}y=4\) and \(\log_{10}x-3\log_{10}y=0\)

   • Let \(u=\log_{10}x\), \(v=\log_{10}y\): \(u+v=4\) and \(u-3v=0\)
   • Subtract: 4v = 4, so v = 1 and u = 3
   • \(x=10^3=1000\), \(y=10\)

2. (b) \(2\log_{10}x-\ln y=1\) and \(\log_{10}x+\ln y^{2}=8\)

   • Let \(u=\log_{10}x\), \(v=\ln y\): \(2u-v=1\) and \(u+2v=8\)
   • From the first, v = 2u − 1. Then u + 4u − 2 = 8, so u = 2 and v = 3
   • \(x=10^2=100\), \(y=e^3\)

In each part a and b are constants with \(a>1\) and \(b>0\). Find the product of all the solutions for x.

1. (a) \(\log_a(x^4)=b\)

   • \(x^4=a^b\), so \(x=a^{b/4}\) or \(x=-a^{b/4}\)
   • Product: \(-a^{b/4}\times a^{b/4}=-a^{b/2}\)

2. (b) \((\log_ax)^2-5\log_ax+6=0\)

   • Let \(L=\log_ax\): \(L^2-5L+6=0\), so \(L=2\) or \(L=3\)
   • \(x=a^2\) or \(x=a^3\)
   • Product: \(a^5\)

Solve \(27^{x} = \dfrac{9^{x+2}}{3}\).

• \(27 = 3^3\), \(9 = 3^2\) and \(3 = 3^1\)
• \(3^{3x} = \dfrac{3^{2(x+2)}}{3^1} = 3^{2x+3}\)
• \(3x = 2x + 3\), so x = 3