Mixed exponent and logarithm practice Problem set A

Solve each equation.

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

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 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 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.

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