Mixed exponent and logarithm practice Problem set B

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

Write each expression as a single logarithm.

1. (a) \(5\ln p+3\ln q\)
2. (b) \(\tfrac14\ln m-2\ln n\)
3. (c) \(\ln20-\ln4+3\ln2\)

Solve each equation.

1. (a) \(\log_{10}(x-7)=3\)
2. (b) \(\log_{10}(4x+1)=2\)
3. (c) \(\ln(x+2)=1.5\)

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

1. (a) \(6^x=45\)
2. (b) \(4\times3^x+5=80\)
3. (c) \(1.8\times1.05^x=36\)

Find the exact value of each logarithm.

1. (a) \(\log_{10}\dfrac{1}{\sqrt[4]{10}}\)
2. (b) \(\log_{10}\sqrt{10^{7}}\)
3. (c) \(\ln\dfrac{e^{4}}{\sqrt[3]{e}}\)

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

1. (a) \(\log_{x}125=3\)
2. (b) \(\log_{x}64=-3\)
3. (c) \(\log_{x}27=\tfrac32\)

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

1. (a) \(8^{x+1}=16^{2x-3}\)
2. (b) \(25^{2x-1}=125^{x+2}\)
3. (c) \(2^{3x+1}=\dfrac1{64}\)

Solve \(5^{2x-1}=3^{x+4}\), giving x in the form \(\dfrac{\ln p}{\ln q}\). Then give x to 3 significant figures.

Solve each equation.

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

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

1. (a) \(7^{x-1} = 343\)
2. (b) \(4^{x} = 32\)
3. (c) \(3^{2x-1} = 81^{x-3}\)

The value of a car is modelled by \(V=24\,000\times0.88^{t}\) euros, where t is the time in years after it is bought.

1. (a) Find the value when it is bought.
2. (b) Find the value after 3 years.
3. (c) Use your calculator to find when the value has fallen to 10 000 euros.

The number of users, in thousands, of two apps after t months is \(P=300\times1.15^{t}\) and \(Q=700\times1.06^{t}\). Find when the two numbers are equal. Give your answer to 3 significant figures.

The number of views of a video doubles every 4 hours. Find how long it takes to multiply by 30. Give your answer in hours to 3 significant figures.

The number of members of an online forum is modelled by \(N=320\times1.35^{t}\), where t is the time in weeks.

1. (a) Write down the number of members at the start, and find the number after 6 weeks. Give your answer to 3 significant figures.
2. (b) Use your calculator to find when there are 5000 members. Give t to 3 significant figures.
3. (c) Write down the percentage increase each week.

The amount of caffeine in a person's blood is modelled by \(C=C_0e^{-0.139t}\), where t is the time in hours after drinking a coffee.

1. (a) Find the half-life of caffeine in the blood. Give your answer to 3 significant figures.
2. (b) Find the time for the amount to fall to 15% of its starting value. Give your answer to 3 significant figures.

The number of views of a video is modelled by \(V=V_0e^{kt}\), where t is the time in days. There are 2500 views at \(t=0\) and 9000 views at \(t=5\).

1. (a) Find k to 3 significant figures.
2. (b) Find when there are 50 000 views. Give your answer to 3 significant figures.

Given that \(9\times6^{x}=24\times3^{x}\):

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

Solve each pair of simultaneous equations.

1. (a) \(\log_{10}x+\log_{10}y=5\) and \(\log_{10}x-4\log_{10}y=0\)
2. (b) \(3\log_{10}x-\ln y=7\) and \(\log_{10}x^{2}+3\ln y=12\)

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) \(\log_a(x^6)=b\)
2. (b) \((\log_ax)^2-7\log_ax+10=0\)

Solve \(8^{x} = \dfrac{4^{x+3}}{2}\).