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Logarithms

Evaluate or simplify a logarithm

Enter a logarithm with its base visible. The calculator looks first for an exact exponent, then uses change of base when a decimal approximation is helpful.

Use the math keyboard or your device keyboard.
Try an example

Result

8 is the third power of 2

log⁡2(8)=3\log_{2}(8)=3
Conditions
  • The argument 8 is positive.
  • The base 2 is positive and is not 1.
  • The result is exact because 2^3=8.

Steps

  1. Write the exponential question Ask which exponent on 2 produces 8.2y=82^y=8
  2. Express the argument as a power Eight is 2 multiplied by itself three times.8=238=2^3
  3. Match the exponents Equal positive-base powers have equal exponents.y=3y=3
Independent check

Substitution into exponential form gives 2^3=8 exactly.

Scope

What this logarithm covers

The value of log_b(x) is the exponent y for which b^y=x. For real logarithms, the argument must be positive, the base must be positive, and the base cannot equal 1.

Evaluate exact logarithms

Recognize when the argument is an integer or fractional power of the stated base.

Examples: log_2(32), log_5(1/25), log_9(3)

Use any valid real base

Work with a positive base other than 1, including integer, fractional, and decimal bases.

Examples: log_3(81), log_(1/2)(8)

Return exact and approximate forms

Keep an exact quotient such as ln(7)/ln(2), then label a decimal only when no simpler exact exponent is available.

Examples: log_2(7), log_3(10)

Simplify logarithmic structure

Apply inverse, product, quotient, and power relationships while keeping all real-domain conditions visible.

Examples: log_b(b^x), log_2(8x)-log_2(x)

How to use it

Enter enough information for one clear task

  1. 1
    Show the base explicitly

    Use log(value, base) or visual subscript notation. If no base is shown, common log means base 10 and ln means base e.

  2. 2
    Choose the intended task

    Evaluate a numerical value, simplify an exact expression, or display the change-of-base form without changing the original meaning.

  3. 3
    Check the real domain

    Every logarithm argument must be positive, and every base must be positive and unequal to 1.

  4. 4
    Verify in exponential form

    Raise the base to the reported value. Exact results should reproduce the argument; decimal results should do so within the stated rounding.

Worked inputs

Examples to try

Use these examples to recognize the method, compare equivalent forms, and check your own work.

Positive integer exponent

Write 32 as a power of 2.

log⁡2(32)\log_2(32)

Expected result

55

Negative exponent

Rewrite 1/25 as 5 to the negative 2.

log⁡5(125)\log_5\left(\frac1{25}\right)

Expected result

−2-2

Fractional exponent

Write 3 as 9 to the one-half power.

log⁡9(3)\log_9(3)

Expected result

12\frac12

Change of base

Divide natural logarithms, then approximate only at the end.

log⁡2(7)\log_2(7)

Expected result

ln⁡7ln⁡2≈2.807355\frac{\ln 7}{\ln 2}\approx2.807355

Base between zero and one

Solve (1/2)^y=8 by comparing powers of 2.

log⁡1/2(8)\log_{1/2}(8)

Expected result

−3-3

Simplify an inverse pair

Log base 3 and the base-3 exponential undo each other for every real exponent.

log⁡3(32x−1)\log_3(3^{2x-1})

Expected result

2x−12x-1

Reject an invalid real argument

No real power of a positive base produces a negative number.

log⁡4(−16)\log_4(-16)

Expected result

undefined over R\text{undefined over }\mathbb{R}

Complete example

Keep a noninteger logarithm exact before rounding

Fourteen is not an integer power of 3, so change of base gives an exact quotient and a controlled approximation.

log⁡3(14)\log_3(14)
  1. 1
    Confirm the domain

    The argument and base are positive, and the base is not 1.

    14>0,3>0,3≠114>0,\qquad 3>0,\qquad 3\ne1
  2. 2
    Apply change of base

    Natural logarithms are convenient, but common logarithms would give the same quotient.

    log⁡3(14)=ln⁡14ln⁡3\log_3(14)=\frac{\ln 14}{\ln 3}
  3. 3
    Approximate at the end

    Evaluate the quotient without rounding the numerator and denominator separately.

    ln⁡14ln⁡3≈2.4021735\frac{\ln 14}{\ln 3}\approx2.4021735
log⁡3(14)=ln⁡14ln⁡3≈2.402174\log_3(14)=\frac{\ln14}{\ln3}\approx2.402174

Verification: Raising 3 to the unrounded quotient returns 14 to numerical precision, and 3^2<14<3^3 confirms that the value lies between 2 and 3.

Avoidable errors

Common mistakes and how to fix them

Using a nonpositive argument

Problem: Treat log_b(0) or log_b(-4) as an ordinary real number.

Why it matters: A positive real base raised to a real exponent is always positive.

Better approach: Require every real logarithm argument to be strictly greater than zero.

Allowing base 1

Problem: Compute log_1(5) by change of base.

Why it matters: Every power of 1 equals 1, and change of base would divide by ln(1)=0.

Better approach: Require b>0 and b not equal to 1 before evaluating log_b(x).

Distributing a logarithm across a sum

Problem: Rewrite log(x+y) as log(x)+log(y).

Why it matters: The product rule applies to multiplication, not addition.

Better approach: Keep log(x+y) intact unless the sum itself can be simplified first.

Rounding too early

Problem: Round ln(14) and ln(3) separately to one decimal place.

Why it matters: The quotient can lose meaningful accuracy even when each rounded input looks close.

Better approach: Keep the exact ratio and round only the final decimal to the requested place.

Trust the result for the right reasons

Checks, assumptions, and limits

How results are checked

  • Exact results are converted back to exponential form.
  • Change-of-base decimals are checked by exponentiating the unrounded value.
  • Argument and base restrictions are enforced before any algebraic simplification.
  • Simplified logarithmic expressions retain conditions on every original argument.

When to stop and revise the input

  • The calculator uses real logarithms and does not select complex logarithm branches.
  • An omitted base is interpreted only through explicit common-log or natural-log notation.
  • Decimal output is an approximation and is labeled with the approximately-equal sign.
  • Logarithmic equations with multiple candidate solutions belong in the general algebra solver.

Common questions

Evaluate or simplify a logarithm FAQ

What does a logarithm calculate?

It calculates an exponent. The statement log_b(x)=y means exactly that b^y=x, subject to the real-domain restrictions on b and x.

What base is log without a subscript?

In many algebra and calculator contexts, log means base 10. Because conventions can vary, the page keeps the interpreted base visible before calculating.

Why is the logarithm argument required to be positive?

A positive real base raised to any real power remains positive, so no real exponent can produce zero or a negative argument.

Is change of base an approximation?

No. The identity log_b(x)=ln(x)/ln(b) is exact for valid real inputs. Only a rounded decimal evaluation of that quotient is approximate.

Can this calculator simplify logarithms as well as evaluate them?

Yes, for supported product, quotient, power, and inverse relationships. Use the dedicated expansion or condensing tool when the direction of the rewrite is the main task.

Sources and curriculum alignment

This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.

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Created by Mathos AI. Methods, conditions, and checks are shown so you can review the mathematical reasoning.