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Math Practice

Common and Natural Log Practice Problems

The common log uses base 10 and the natural log uses base e, but both follow the same definition. These twelve problems practise evaluating them, cancelling ln against e, and undoing a log in an equation by converting to exponential form.

  1. 1.

    Evaluate .

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    With no base written, means base 10. , so .

  2. 2.

    Simplify .

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    undoes : , so .

  3. 3.

    Simplify .

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    undoes : , so .

  4. 4.

    Evaluate .

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    , so .

  5. 5.

    Solve . Give to three decimal places.

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    The base of is , so .

  6. 6.

    Solve . Give to two decimal places.

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    The base is 10, so .

  7. 7.

    Solve . Give to four decimal places.

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    Take the natural log of both sides: .

  8. 8.

    Is positive or negative? Answer "positive" or "negative".

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    0.3 is between 0 and 1, so must be raised to a negative power to reach it. .

  9. 9.

    Solve . Give to four decimal places.

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    Convert: . Then , so . The argument is positive, so the answer stands.

  10. 10.

    Solve . Give to four decimal places.

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    . A negative log just means is between 0 and 1.

  11. 11.

    Solve .

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    , so and . The negative answer is valid because the argument, , is positive.

  12. 12.

    Does exist as a real number? Answer "yes" or "no".

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    No. would need , but is positive for every real . The natural log is defined only for positive inputs.
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MPhil in Mathematics, Founder of Mathovia
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Common and Natural Log Practice Problems

These problems follow the common and natural logarithms lesson. Remember that alone means base 10 and means base .

The two definitions

Natural log rules with e

RuleStatement
log of 1
log of the base
undoes
undoes

Always check the argument

After solving an equation containing or , substitute your answer back and make sure every argument is positive. A negative answer can be valid; a negative argument never is.

Common mistakes to watch for

  • Mixing up the bases. is base 10; is base .
  • Wrong calculator key. Use LOG for base 10 and LN for base .
  • Negative inputs. Neither is defined for zero or negative numbers.
  • Forgetting the inverse pairs. and .

Frequently Asked Questions

What is the difference between log and ln?+

log with no base means base 10, the common logarithm. ln means base e, about 2.71828, the natural logarithm. They give different values for the same input: log 10 = 1 while ln 10 ≈ 2.3026.

How do I get rid of ln in an equation?+

Raise e to both sides, or equivalently convert to exponential form. ln x = 4 becomes x = e⁴. To get rid of e instead, take ln of both sides.

Can the natural log be negative?+

Yes, whenever the input is between 0 and 1. The input itself must be positive: ln 0 and the log of any negative number are undefined.

How do I enter e² or 10^2.5 as an answer?+

Either type the decimal, rounded as the question asks, or type the expression with a caret, such as e^2 or 10^2.5. Both are checked by value.

Why is ln 1 equal to 0?+

Because e⁰ = 1. The same is true for every base — log 1 = 0 as well — which is why every logarithm graph passes through (1, 0).

More practice problems

Logarithm Practice Problems Logarithm Rules Practice Problems Solving Logarithmic Equations Practice Problems