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IB Mathematics: Applications and Interpretation SL · 12.6: Exponents & Logarithms · 12.6.3

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Mixed exponent and logarithm practice Problem set B

Show your working. Use the laws of logarithms for exact answers and the calculator for decimals. Give 3 significant figures unless told otherwise.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. Solve each equation.

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

  2. 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) \(2^{3x+1}=\dfrac1{64}\)

  3. Solve each equation.

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

  4. 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) \(3^{2x-1} = 81^{x-3}\)

In context

  1. 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.

  2. 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.

Stretch

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

jmaths.xyz · AI SL · 12.6.3 Mixed exponent and logarithm practice