Mixed exponent and logarithm practice Problem set A

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) \(\log_{10}(a^3b)\)
2. (b) \(\log_{10}\!\left(\dfrac{b^2}{ac^4}\right)\)
3. (c) \(\log_{10}\!\left(\dfrac{1000\sqrt b}{a^2}\right)\)

Write each expression as a single logarithm.

1. (a) \(4\ln a+2\ln b\)
2. (b) \(\tfrac12\ln x-3\ln y\)
3. (c) \(\ln12-\ln4+2\ln3\)

Solve each equation.

1. (a) \(\log_{10}(x+5)=2\)
2. (b) \(\log_{10}(3x-2)=1\)
3. (c) \(\ln(x-1)=2\)

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

1. (a) \(7^x=30\)
2. (b) \(3\times5^x+4=50\)
3. (c) \(2.5\times1.07^x=40\)

Find the exact value of each logarithm.

1. (a) \(\log_{10}\dfrac{1}{\sqrt{1000}}\)
2. (b) \(\log_{10}\sqrt[3]{100}\)
3. (c) \(\ln\!\left(e^{2}\sqrt{e}\right)\)

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

1. (a) \(\log_{x}81=4\)
2. (b) \(\log_{x}49=-2\)
3. (c) \(\log_{x}8=\tfrac32\)

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

1. (a) \(4^{2x-1}=32^{x+2}\)
2. (b) \(27^{x+1}=9^{2x-3}\)
3. (c) \(3^{2x-1}=\dfrac1{81}\)

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

Solve each equation.

1. (a) \(\log_{10}(4x) = 2\)
2. (b) \(\log_{10}(x-7) = -1\)
3. (c) \(\ln(3x-2) = 2.5\)

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

1. (a) \(5^{x-2} = 125\)
2. (b) \(9^{x} = 27\)
3. (c) \(2^{3x+1} = 16^{x-3}\)

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.
2. (b) Find the mass after 3 hours.
3. (c) Use your calculator to find when the mass has fallen to 20 mg.

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.

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.

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.
2. (b) Use your calculator to find when there are 10 000 bacteria. Give t to 3 significant figures.
3. (c) Write down the percentage increase each hour.

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

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

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

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

Solve each pair of simultaneous equations.

1. (a) \(\log_{10}x+\log_{10}y=4\) and \(\log_{10}x-3\log_{10}y=0\)
2. (b) \(2\log_{10}x-\ln y=1\) and \(\log_{10}x+\ln y^{2}=8\)

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) \(\log_a(x^4)=b\)
2. (b) \((\log_ax)^2-5\log_ax+6=0\)

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