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AI SL · 12.6

Exponents & Logarithms

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12.6.2

Logarithms and solving exponential and log equations

12.6.2

Logarithms and solving exponential and log equations

Target: I can use log₁₀ and ln as the inverses of 10^x and e^x, and evaluate them with the calculator. (+2 more)

By the end of this section

Key points

Logarithms are inverses

  • \(\log_{10} b = x\) means \(b = 10^x\). \(\ln b = x\) means \(b = e^x\). The number e is about 2.718.
  • \(\log x\) and \(10^x\) undo each other. So do \(\ln x\) and \(e^x\). \(\ln(e^x) = x\) and \(e^{\ln x} = x\).
  • You can only take the log of a positive number. After you solve, check every answer in the original equation.
  • To solve a log equation, get the log on its own, then write it as a power. \(3 + 2\ln x = 1\) gives \(\ln x = -1\), so \(x = e^{-1}\).
  • On a log scale, each step of 1 multiplies the quantity by 10. A difference of \(\Delta\) on the scale means a ratio of \(10^{\Delta}\).

Read

Notes, worked examples and exam style questions.

On the calculator

Evaluating log and ln

  • Use the calculator for log of a number, and ln of a number, for example −log of 3 × 10⁻⁵.
  • Put a bracket round the whole argument of the log.
  • ln is base e. log with no base is base 10.

Screenshot and video to come

Solving an equation numerically

  • Use the solver to find x when the equation has x on both sides, for example 5^(x+2) = 3^(2x−1).
  • Give 3 significant figures unless the question asks for an exact answer.

Screenshot and video to come

Checking with a graph

  • Graph the left side and the right side as two functions.
  • The x-coordinate of the intersection is the solution. Compare it with your exact answer as a decimal.

Screenshot and video to come

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Undoing e to the power x

\(\text{Solve } e^x = 0.2. \text{ Give x to 3 significant figures.}\)

Working
  1. Take ln of both sides

    \(\ln\left(e^x\right) = \ln 0.2\)

    ln undoes e^x. In general ln(e^x) = x, and the same for log and 10^x.

  2. Calculator

    From the calculator: \(x = \ln 0.2 = -1.61\)

    Give 3 significant figures.

Video walkthrough coming
Converting a log equation

\(\text{Find the exact value of } y \text{ if } \log_{10}(y-2) = -2.\)

Working
  1. Convert

    \(y - 2 = 10^{-2}\)

    log₁₀ b = x means b = 10^x.

  2. Evaluate

    \(y - 2 = \frac{1}{100}\)

    A negative index is a reciprocal.

  3. Solve

    \(y = 2.01\)

    Add 2 to both sides.

Video walkthrough coming
Isolate ln, then use e

\(\text{Solve } 3 + 2\ln x = 1. \text{ Give an exact answer, and x to 3 significant figures.}\)

Working
  1. Isolate ln x

    \(2\ln x = -2 \Rightarrow \ln x = -1\)

    Subtract 3, then divide by 2.

  2. Use e

    \(x = e^{-1}\)

    ln x = k means x = e^k. This is the exact answer.

  3. Decimal

    From the calculator: \(x = 0.368\)

    Give 3 significant figures.

Video walkthrough coming
How many times greater?

The energy E, in joules, released by an earthquake of magnitude M satisfies \(\log_{10}E=4.8+1.5M\). How many times more energy does a magnitude 7 earthquake release than a magnitude 5 earthquake?

Working
  1. Difference in the log

    \(\log_{10}E_7-\log_{10}E_5=1.5\times7-1.5\times5=3\)

    The 4.8 cancels.

  2. Use the log law

    \(\log_{10}\dfrac{E_7}{E_5}=3\)

    A difference of logs is the log of the ratio.

  3. Convert

    \(\dfrac{E_7}{E_5}=10^{3}=1000\)

    A magnitude 7 earthquake releases 1000 times the energy.

Video walkthrough coming