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A. Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Exponential and Logarithmic Functions as Inverses

A2.2 A1.1About 70 minutes
Learning goals
  • I can explain why y = log_b x and y = bˣ are inverse functions.
  • I can sketch the graph of y = log_b x by reflecting y = bˣ in the line y = x.
  • I can find the equation of the inverse of an exponential or logarithmic function, including transformed ones.
  • I can state how the domain, range and asymptote of a function and its inverse are related.

Words to know

inverse function
The function f⁻¹ that undoes f: if f(a) = b, then f⁻¹(b) = a. Its graph is the reflection of the graph of f in the line y = x.
f⁻¹(x)
Notation for the inverse of f. The −1 is not an exponent: f⁻¹(x) does not mean 1/f(x).
line of reflection
The line y = x. Reflecting a point (a, b) in it gives (b, a).
one-to-one
A function where each y-value comes from exactly one x-value. Exponential functions are one-to-one, so their inverses are functions too.

Undo, swap, reflect

Remember from Grade 11: in Functions and Equivalent Expressions (MCR3U) you learned three ways to think about an inverse: it undoes the function, it swaps the x- and y-coordinates of every point, and its graph is the reflection of the original in the line y = x.

The exponential function f(x) = 2ˣ takes an exponent and gives a power: f(3) = 8. Its inverse must take the power and give back the exponent: f⁻¹(8) = 3. That is exactly what log₂ does. So:

If f(x) = bˣ, then f⁻¹(x) = log_b x.

And since "inverse" works both ways, the inverse of y = log_b x is y = bˣ.

x−2−10123
y = 2ˣ1/41/21248
Swap the rows of this table and you get a table for y = log₂ x: x = 1/4, 1/2, 1, 2, 4, 8 and y = −2, −1, 0, 1, 2, 3.
xy−4−3−2−1123456−4−3−2−1123456(0, 1)(1, 0)(1, 2)(2, 1)(2, 4)(4, 2)y = 2ˣy = log₂ xy = x
y = 2ˣ and y = log₂ x are mirror images in the line y = x. Every point (a, b) on one graph matches the point (b, a) on the other, such as (1, 2) and (2, 1).

What swaps and what stays

  • Domain and range swap. y = 2ˣ has domain {x ∈ ℝ} and range {y ∈ ℝ | y > 0}. y = log₂ x has domain {x ∈ ℝ | x > 0} and range {y ∈ ℝ}.
  • Intercepts swap. y-intercept 1 becomes x-intercept 1.
  • Asymptotes swap. The horizontal asymptote y = 0 becomes the vertical asymptote x = 0.
  • Increasing stays increasing (for b > 1), and decreasing stays decreasing (for 0 < b < 1).
Worked example

Example 1: Find and check an inverse

Find the inverse of f(x) = 5ˣ. Then show that f⁻¹(f(2)) = 2 and f(f⁻¹(125)) = 125.

  1. Write y = 5ˣ. Swap x and y: x = 5ʸ.
  2. Solve for y by rewriting in logarithmic form: y = log₅ x. So f⁻¹(x) = log₅ x.
  3. f(2) = 25, and f⁻¹(25) = log₅ 25 = 2 ✓
  4. f⁻¹(125) = log₅ 125 = 3, and f(3) = 5³ = 125 ✓

f⁻¹(x) = log₅ x. Applying a function and then its inverse returns the starting value. You will study this more in the unit Combining Functions and Modelling.

Worked example

Example 2: Inverse of a transformed exponential function

Find the inverse of f(x) = 2^(x − 1) + 3. State the domain, range and asymptote of f and of f⁻¹.

  1. y = 2^(x − 1) + 3. Swap: x = 2^(y − 1) + 3.
  2. Isolate the power: x − 3 = 2^(y − 1).
  3. Logarithmic form: y − 1 = log₂ (x − 3).
  4. So f⁻¹(x) = log₂ (x − 3) + 1.
  5. f: domain {x ∈ ℝ}, range {y ∈ ℝ | y > 3}, horizontal asymptote y = 3.
  6. f⁻¹: domain {x ∈ ℝ | x > 3}, range {y ∈ ℝ}, vertical asymptote x = 3.

f⁻¹(x) = log₂ (x − 3) + 1. The domain and range swap, and the asymptote y = 3 becomes x = 3.

xy−3−2−112345678910−3−2−112345678910(1, 4)(2, 5)(3, 7)(4, 1)(5, 2)(7, 3)y = 3x = 3f(x) = 2^(x − 1) + 3f⁻¹(x) = log₂ (x − 3) + 1
f(x) = 2^(x − 1) + 3 and its inverse are reflections in y = x. The point (1, 4) on f matches (4, 1) on f⁻¹, and the asymptote y = 3 becomes x = 3.
Worked example

Example 3: Inverse of a logarithmic function

Find the inverse of g(x) = log₃ (x + 2) − 4.

  1. y = log₃ (x + 2) − 4. Swap: x = log₃ (y + 2) − 4.
  2. Isolate the log: x + 4 = log₃ (y + 2).
  3. Exponential form: y + 2 = 3^(x + 4).
  4. So g⁻¹(x) = 3^(x + 4) − 2.
  5. Check with a point: g(7) = log₃ 9 − 4 = −2. g⁻¹(−2) = 3² − 2 = 7 ✓

g⁻¹(x) = 3^(x + 4) − 2

A quick check

Pick a convenient point on the original function, swap its coordinates, and make sure the swapped point satisfies your inverse. Also check that the asymptotes swapped: a horizontal asymptote y = k must become the vertical asymptote x = k.

Just for fun

An exponential function and its logarithm are like a pair of mittens: the same shape, mirror images of each other, and not much use without the other one when you're solving equations in January.

Hands-on activity

Fold to find the inverse

You need: a square of graph paper (or draw a grid on plain paper), a pencil, a ruler and a bright window.

  1. Draw axes with the origin near one corner so you have room in all directions, using 1 cm per unit. Draw the line y = x lightly.
  2. Plot y = 2ˣ for x = −3, −2, −1, 0, 1, 2, 3 (y = 1/8, 1/4, 1/2, 1, 2, 4, 8; skip 8 if it runs off the paper) and join the points with a smooth curve. Darken the curve.
  3. Fold the paper exactly along the line y = x. Hold it against the window and trace the curve onto the other half.
  4. Unfold. Read three points from the traced curve and check that they satisfy y = log₂ x.

What to look for: the traced curve passes through (1, 0), (2, 1) and (4, 2), and hugs the y-axis just as the original hugs the x-axis. Folding along y = x swaps the x- and y-coordinates of every point, which is exactly what an inverse does.

Your turn

1

Find the inverse of f(x) = 7ˣ.

2

Find the inverse of g(x) = log x.

3

The point (3, 27) is on y = 3ˣ. Which point must be on y = log₃ x?

4

State the domain and range of y = log₆ x, using what you know about y = 6ˣ.

5

Find the inverse of f(x) = 2ˣ − 5.

6

Find the inverse of h(x) = log₄ x + 1.

7

Without a calculator, evaluate log₂ (2^(log₂ 8)).

More practice: warm-up

1

What line do you reflect in to get the graph of an inverse?

2

What is the inverse of y = (1/2)ˣ?

3

The asymptote of y = 4ˣ is y = 0. What is the asymptote of y = log₄ x?

4

If f(x) = 9ˣ, what is f⁻¹(81)?

5

True or false: f⁻¹(x) means 1/f(x).

More practice: on your own

1

Find the inverse of f(x) = 3^(x + 2).

2

Find the inverse of f(x) = 5(2ˣ).

3

Find the inverse of g(x) = log₂ (x − 1) + 3 and state its range.

4

f(x) = 10^(x − 2) + 4. State the equation of the asymptote of f⁻¹ and the domain of f⁻¹.

5

Show that f(x) = 2ˣ and g(x) = log₂ x are inverses by evaluating f(g(16)) and g(f(−3)).

Challenge

1

Find the inverse of f(x) = 3(2^(x − 1)) + 5 and state its domain.

2

Explain why y = 2ˣ and y = log₂ x never intersect, even though they are reflections in y = x.

3

Does y = (1/16)ˣ meet its inverse y = log_(1/16) x? Check x = 1/2 and x = 1/4.

Check your understanding

Quiz

1.

What is the inverse of f(x) = 6ˣ?

2.

The point (2, 100) is on y = 10ˣ. Which point is on y = log x?

3.

In which line are the graphs of y = 3ˣ and y = log₃ x reflections of each other?

4.

What is the inverse of f(x) = 2ˣ + 1?

5.

f(x) = 5ˣ − 2 has the horizontal asymptote y = −2. What is the asymptote of f⁻¹?

6.

What is the range of the inverse of g(x) = log₇ x?

For parents and teachers

Big idea (A2.2): y = bˣ and y = log_b x are inverse functions: each undoes the other, their graphs are reflections in y = x, and their domains, ranges, intercepts and asymptotes swap.

How to help: Have your teen sketch y = 2ˣ, draw y = x, and then sketch the mirror image, saying aloud which features swap. For an equation like f(x) = 2^(x − 1) + 3, ask them to talk through the steps: swap, isolate the power, switch to log form.

Common difficulties:

  • Reading f⁻¹(x) as 1/f(x).
  • Swapping x and y but then solving for x instead of y.
  • Forgetting to isolate the power (or the log) before switching forms.
  • Not swapping the asymptote: a horizontal asymptote y = k becomes the vertical line x = k.