Mixed exponent and logarithm practice Problem set A Worked solutions

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

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 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 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%.

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