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Graphing Logarithmic Functions Practice Problems

A logarithmic graph is fixed by its vertical asymptote and a couple of points. These twelve problems practise reading those features from the equation: solving 'argument > 0' for the domain, placing the asymptote, and finding the intercepts after shifts and reflections.

  1. 1.

    What is the domain of ? Use interval notation.

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    The argument must be positive: , so . Domain .

  2. 2.

    What is the vertical asymptote of ? Give the value of .

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    The asymptote is where the argument equals zero: , so .

  3. 3.

    What is the x-coordinate of the x-intercept of ?

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    when . Every parent log graph crosses the x-axis at .

  4. 4.

    What is the range of ? Use interval notation.

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    A logarithm can output any real number, so the range is . It is the domain that is restricted.

  5. 5.

    What is the domain of ? Use interval notation.

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    gives , so . Domain .

  6. 6.

    What is the x-coordinate of the x-intercept of ?

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    is zero when its argument is 1: , so .

  7. 7.

    What is the x-coordinate of the x-intercept of ?

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    Set : , so .

  8. 8.

    The graph of passes through . Find .

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    means , so . The base must be positive, so is rejected.

  9. 9.

    What is the x-coordinate of the x-intercept of ?

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    , so and .

  10. 10.

    What is the y-coordinate of the y-intercept of ? Give it to three decimal places.

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    .

  11. 11.

    What is the x-coordinate of the x-intercept of ?

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    , so and .

  12. 12.

    What is the domain of ? Use interval notation.

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    gives . The domain is and the vertical asymptote is .
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Graphing Logarithmic Functions Practice Problems

The graphing logarithmic functions lesson explains each feature used here, including the natural log graph.

Key features of

FeatureValue
Domain
Range
x-intercept
Vertical asymptote
Other easy points and

After a transformation

For any intercept, isolate the logarithm and convert to exponential form. For the domain, solve “argument ” — this also works when the argument is something like , where the domain lies to the left of the asymptote.

Common mistakes to watch for

  • Putting the asymptote in the wrong place. has its vertical asymptote at .
  • Shifting the wrong way. moves the graph 2 left.
  • Wrong domain. The domain is where the inside is positive.
  • Forgetting key points. Every passes through and .

Frequently Asked Questions

How do I find the domain of a logarithmic function?+

Set the argument greater than zero and solve the inequality. The boundary value you get is also the vertical asymptote.

Which transformation moves the asymptote of a log graph?+

The horizontal shift. In y = log_b(x − h) + k the asymptote is x = h; the vertical shift k moves the graph up or down but leaves the vertical asymptote where it is.

Why does every parent log graph pass through (1, 0)?+

Because log_b 1 = 0 for every base — any base to the power 0 is 1. After a shift, the x-intercept is wherever the whole log expression equals zero.

What is the range of a logarithmic function?+

All real numbers. A log graph rises (or falls) without bound in both directions, just very slowly on one side. It has no horizontal asymptote.

How can I find the base from a graph?+

Use any marked point (x, y) and convert log_b x = y to bʸ = x. A point with y = 1 is ideal: it sits at x = b.

More practice problems

Graphing Exponential Functions Practice Problems Common and Natural Log Practice Problems Logarithm Practice Problems