Logarithmic Functions

The exponential function y = 2ˣ turns the number 3 into the number 8. What if we want to go the other way, from 8 back to 3? We need a function that answers "what power do you have to raise 2 to in order to get 8". That is the logarithm: log₂ 8 = 3.

The function y = logₐ x is called a logarithmic function. It is the inverse function of the exponential function y = aˣ: it does exactly the opposite job.

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Contents

• The graph as a mirror image
• Properties of the logarithmic function
• Base greater than 1 and less than 1
• Shifted graphs
• The inverse of a shifted exponential
• Common mistakes
• Summary
• Try it yourself
• Practice

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The graph as a mirror image

An inverse function swaps the input and the output. The point (3, 8) lies on the graph of y = 2ˣ, so the point (8, 3) lies on the graph of y = log₂ x. In the same way (0, 1) ↔ (1, 0) and (1, 2) ↔ (2, 1).

Swapping the coordinates is a reflection in the line y = x.

The graphs of y = aˣ and y = logₐ x are symmetric about the line y = x.

More about inverse functions in the article One-to-One and Inverse Functions.

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Properties of the logarithmic function

When x and y swap, the domain and the range swap too:

• The domain is the positive numbers only: neither log₂ 0 nor log₂ (−4) exists.
• The range is all real numbers. A logarithm can be negative too: log₂ ½ = −1.
• The graph always passes through the point (1, 0), because logₐ 1 = 0, and through the point (a, 1), because logₐ a = 1.
• The asymptote is the y-axis: as x gets closer to zero, the logarithm decreases (or increases) without limit.

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Base greater than 1 and less than 1

• For a > 1 the logarithmic function is increasing (for example log₂ x: 1 → 0, 2 → 1, 4 → 2, 8 → 3). It grows very slowly, though.
• For 0 < a < 1 it is decreasing (for example log base ½: 2 → −1, 4 → −2).

The graphs of log₂ x and log base ½ of x are symmetric about the x-axis.

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Shifted graphs

Just as with other functions:

• y = logₐ (x − m) is the graph shifted m to the right, and the vertical asymptote is x = m,
• y = logₐ x + n is the graph shifted n up.

Worked example 1

y = log₂ (x − 1) + 2

1. Asymptote: x = 1. Domain: x > 1.
2. The point (1, 0) moves to (2, 2), where the argument is x − 1 = 1.
3. The base is 2 > 1, so the function is increasing.

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The inverse of a shifted exponential

The method is the same as for any inverse function: swap x and y, then make y the subject.

Worked example 2

Find the inverse function of f(x) = 2ˣ + 1.

1. y = 2ˣ + 1.
2. Swap: x = 2ʸ + 1.
3. 2ʸ = x − 1, so y = log₂ (x − 1).

Result: f⁻¹(x) = log₂ (x − 1).

Check the domain and range: f takes values greater than 1, so the domain of f⁻¹ is x > 1. That fits, because the logarithm of x − 1 needs x > 1.

💡 In general: the inverse of y = a^(x − m) + n is y = logₐ (x − n) + m. The shifts swap, just like the coordinates.

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Common mistakes

• "log₂ x is defined for all x." No, only for x > 0.
• Mixing up the asymptotes. The exponential has a horizontal asymptote (the x-axis), the logarithm a vertical one (the y-axis).
• The wrong shift in the inverse function. The inverse of y = 2ˣ + 1 is not y = log₂ x − 1 but y = log₂ (x − 1).
• Reflecting in an axis. The graphs of aˣ and logₐ x are symmetric about the line y = x, not about the y-axis.

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Summary

• y = logₐ x is the inverse function of y = aˣ.
• Domain (0, ∞), range ℝ, the point (1, 0), the asymptote x = 0.
• For a > 1 it is increasing, for 0 < a < 1 it is decreasing.
• The graph is the mirror image of the graph of aˣ in the line y = x.
• The inverse of y = a^(x − m) + n is y = logₐ (x − n) + m.

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Try it yourself

1. What is the domain of y = log₃ (x + 2)?
2. Which point does the graph of y = log₅ x pass through where y = 1?
3. The point (2, 9) lies on the graph of y = 3ˣ. Which point lies on the graph of y = log₃ x?
4. Find the inverse function of f(x) = 3^(x − 2).

Answers:

1. x + 2 > 0, so x > −2.
2. (5, 1).
3. (9, 2).
4. x = 3^(y − 2), y − 2 = log₃ x, f⁻¹(x) = log₃ x + 2.

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Practice

• 📊 Logarithmic function and the graph of the inverse
• 📊 Graph and properties of the exponential function

👉 Back to the start: Exponential Functions