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Construct exponential and logarithmic inverse pairs

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Inverse exponential and logarithmic functions

An exponential function and a logarithm with the same valid base are inverse functions. Their graphs reflect in the diagonal line, so coordinates, domain and range, and asymptote directions are exchanged

\[f(x)=a^{x}\quad\Longleftrightarrow\quad f^{-1}(x)=\log_{a}\left(x\right)\]
  1. 1Interchange input and output To find an inverse rule, interchange x and y before isolating the new output \(\displaystyle y=f(x)\quad\Longrightarrow\quad x=f(y)\)
  2. 2Swap corresponding points A point on a function and its inverse has its coordinates in the opposite order \(\displaystyle P=(p,q)\quad\Longrightarrow\quad P'=\left(q,p\right)\)
  3. 3Exchange domain and range The domain of the inverse is the range of the original function, and the range of the inverse is the original domain \(\displaystyle \operatorname{Dom}(f^{-1})=\operatorname{Ran}(f)\)
  4. 4Reflect asymptotes Reflection in y = x changes a horizontal asymptote into a vertical asymptote and a vertical asymptote into a horizontal one \(\displaystyle y=k\quad\Longleftrightarrow\quad x=k\)
Year 12, VCE Units 3 & 4, Chapter 4, 4 difficulty levels, No calculatorBuild a worksheet

Inverse exponential and logarithmic functions

An exponential function and a logarithm with the same valid base are inverse functions. Their graphs reflect in the diagonal line, so coordinates, domain and range, and asymptote directions are exchanged

\[f(x)=a^{x}\quad\Longleftrightarrow\quad f^{-1}(x)=\log_{a}\left(x\right)\]
  1. Interchange input and output To find an inverse rule, interchange x and y before isolating the new output \(\displaystyle y=f(x)\quad\Longrightarrow\quad x=f(y)\)
  2. Swap corresponding points A point on a function and its inverse has its coordinates in the opposite order \(\displaystyle P=(p,q)\quad\Longrightarrow\quad P'=\left(q,p\right)\)
  3. Exchange domain and range The domain of the inverse is the range of the original function, and the range of the inverse is the original domain \(\displaystyle \operatorname{Dom}(f^{-1})=\operatorname{Ran}(f)\)
  4. Reflect asymptotes Reflection in y = x changes a horizontal asymptote into a vertical asymptote and a vertical asymptote into a horizontal one \(\displaystyle y=k\quad\Longleftrightarrow\quad x=k\)

Worked example

Find the inverse rule of the function shown

\[f(x)=2^{x-1}+4\]
  1. \(\displaystyle y=2^{x-1}+4\)Write the function rule with y as the output
  2. \(\displaystyle x=2^{y-1}+4\)Interchange x and y to reverse the input-output relationship
  3. \(\displaystyle x-4=2^{y-1}\)Subtract 4 so the exponential term is isolated
  4. \(\displaystyle y-1=\log_{2}\left(x-4\right)\quad\Longrightarrow\quad f^{-1}(x)=\log_{2}\left(x-4\right)+1\)Use exponential-logarithm equivalence and add 1 to isolate y

Common mistakes

Using a reciprocal instead of an inverse
✕\(\displaystyle f(x)=2^{x}\quad\Rightarrow\quad f^{-1}(x)=\frac{1}{2^{x}}\)✓\(\displaystyle f(x)=2^{x}\quad\Rightarrow\quad f^{-1}(x)=\log_{2}\left(x\right)\)
A reciprocal changes the output value, while an inverse reverses the input and output relationship
Not swapping point coordinates
✕\(\displaystyle P=(2,8)\quad\Longrightarrow\quad P'=(2,8)\)✓\(\displaystyle P=(2,8)\quad\Longrightarrow\quad P'=(8,2)\)
Reflection in the diagonal line exchanges the x- and y-coordinates of every corresponding point
Not swapping domain and range
✕\(\displaystyle \operatorname{Dom}\left(f^{-1}\right)=\operatorname{Dom}\left(f\right)\)✓\(\displaystyle \operatorname{Dom}\left(f^{-1}\right)=\operatorname{Ran}\left(f\right)\)
The inverse uses the original range as its domain and the original domain as its range

Difficulty levels

introA basic same-base inverse is formed and a point is reflected by swapping coordinates
standardOne translated exponential or logarithmic member is inverted with its domain and range exchanged
challengeA rule with two translations is inverted and its mapping features are tracked
examThe only matching inverse rule is selected and its exact asymptote and intercept features are recorded

Questions about this topic

How is an inverse different from a reciprocal?+

An inverse reverses the input-output relationship, while a reciprocal replaces a non-zero value by its multiplicative reciprocal

Why do inverse points swap coordinates?+

If a function maps an input to an output, its inverse maps that output back to the input, so the point’s coordinates swap places

What happens to an asymptote under inversion?+

Reflection in the diagonal line swaps horizontal and vertical directions while retaining the asymptote value

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 4: Logarithmic functions

In this chapter

Connect exponential and logarithmic formsApply logarithm laws and inverse identitiesEvaluate logarithms and antilogarithmsExpress one logarithmic quantity in terms of anotherUse a formula-bearing logarithmic scaleSolve exponential equations and inequalities exactlySolve exponential and logarithmic equationsSolve literal logarithmic and indicial equationsAnalyse exponential and logarithmic graphsConstruct exponential and logarithmic inverse pairsInterpret and use an exponential modelInterpret and use a logarithmic modelDetermine and use a logarithmic model