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AI SL · 12.6

Exponents & Logarithms

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12.6.3

Mixed exponent and logarithm practice

12.6.3

Mixed exponent and logarithm practice

Target: I can solve equations such as log(x + 5) = 2 by rewriting them as powers of 10 or e. (+1 more)

By the end of this section

Key points

Logs undo powers

  • \(10^x\) and \(\log_{10}x\) undo each other. So do \(e^x\) and \(\ln x\). \(\log_{10}A=c\) means \(A=10^c\).
  • Integer exponents: \(a^m\times a^n=a^{m+n}\), \((a^m)^n=a^{mn}\), \(a^{-n}=\dfrac1{a^n}\).
  • If both sides can be written as powers of the same base, make the exponents equal.
  • Check a log answer: the number inside the log must be positive.

Exponential models

  • At the start t = 0, so \(a\times b^{0}=a\).
  • To find when a model reaches a value, put the value in and solve for t. Use the calculator's solver, or logs.
  • A quantity that doubles every T units is multiplied by \(2^{t/T}\) after t units.
  • Give the answer with its units and decide whether it makes sense.

On the calculator

log and ln values

  • Use the log and ln keys to evaluate a logarithm.
  • For log base 10, use log. For base e, use ln.
  • Round at the end, not in the middle.

Screenshot and video to come

Solving an exponential equation

  • Open the numerical solver and type the equation with t as the unknown.
  • Read the value of t and give it to 3 significant figures.
  • Check by putting the value back in the model.

Screenshot and video to come

Evaluating ln p / ln q

  • Type the whole fraction with brackets: ln(50) / ln(1.6).
  • Do not round ln p or ln q before dividing.

Screenshot and video to come

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Log of a linear expression

Solve \(\log_{10}(2x+1)=2\).

Working
  1. Undo the log

    \(2x+1=10^{2}\)

    \(\log_{10}A=2\) means \(A=10^2\).

  2. Solve

    \(2x=99\), so \(x=49.5\)

    Subtract 1, then divide by 2.

  3. Check

    \(\log_{10}(2\times49.5+1)=\log_{10}100=2\)

    The check puts x back in.

Video walkthrough coming
Time to reach a value

The number of followers of a channel is modelled by \(F=1200\times1.08^{t}\), where t is the time in months. Use the calculator to find when F reaches 3000.

Working
  1. Write the equation

    \(1200\times1.08^{t}=3000\)

    Put F = 3000.

  2. Solve for t

    Solve for t: \(t=11.9\)

    Use the calculator's numerical solver. Give 3 significant figures.

  3. Check

    \(1200\times1.08^{11.9}\approx3000\)

    State the answer with units: months.

Video walkthrough coming