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Logarithm properties

Expand a logarithmic expression

Enter one logarithm whose argument contains products, quotients, or powers. The result separates that structure without inventing a rule for addition.

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Try an example

Result

Separate the quotient and product, then bring exponents forward

2log⁡x+log⁡y−log⁡z2\log x+\log y-\log z
Conditions
  • The displayed form assumes x>0, y>0, and z>0.
  • The selected base 10 is positive and is not 1.
  • No rule is applied across addition inside an argument.

Steps

  1. Expand the quotient A logarithm of a quotient becomes numerator log minus denominator log.log⁡(x2y)−log⁡z\log(x^2y)-\log z
  2. Expand the numerator product The product becomes a sum of two logarithms.log⁡(x2)+log⁡y−log⁡z\log(x^2)+\log y-\log z
  3. Move powers to coefficients Apply the power rule to each positive variable.2log⁡x+log⁡y−log⁡z2\log x+\log y-\log z
Independent check

Condensing the expanded result reverses each property and returns log of x squared times y divided by z under the same conditions.

Scope

What this logarithm expansion covers

To expand logarithms, turn products into sums, quotients into differences, and powers into coefficients. Each separated logarithm still needs a positive argument, and sums inside an argument do not split.

Separate products

Turn the logarithm of a positive product into the sum of logarithms of its positive factors.

Examples: log_b(xy), ln(6x)

Separate quotients

Turn a positive quotient into numerator log minus denominator log without losing denominator restrictions.

Examples: log_b(x/y), ln((x+1)/3)

Bring powers forward

Rewrite an exponent as a multiplier, including rational exponents used for roots.

Examples: log_b(x^4), ln(sqrt(x))

Preserve unsplittable sums

Leave addition and subtraction inside a logarithm intact because no sum or difference property exists.

Examples: log_b(x+3), ln(x^2+1)

How to use it

Enter enough information for one clear task

  1. 1
    Enter one complete logarithm

    Keep the base visible and use parentheses around the entire argument, especially when it contains a quotient.

  2. 2
    State sign assumptions

    Choose positive-variable assumptions only when the problem grants them; otherwise use absolute values where the identity requires them.

  3. 3
    Work from outer structure inward

    Separate a quotient first, then products, then powers. This order makes signs and coefficients easier to track.

  4. 4
    Reverse the steps to verify

    Condense the final sum and difference under the same conditions and confirm that it reconstructs the original logarithm.

Worked inputs

Examples to try

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

Two-factor product

Use the product rule under x>0.

log⁡2(3x)\log_2(3x)

Expected result

log⁡23+log⁡2x\log_2 3+\log_2 x

Simple quotient

Subtract the denominator logarithm under x>0.

ln⁡(x5)\ln\left(\frac{x}{5}\right)

Expected result

ln⁡x−ln⁡5\ln x-\ln5

Integer power

Bring the exponent forward for positive x.

log⁡7(x4)\log_7(x^4)

Expected result

4log⁡7x4\log_7x

Square root

Rewrite the root as x to the one-half power.

ln⁡x\ln\sqrt{x}

Expected result

12ln⁡x,x>0\frac12\ln x,\quad x>0

Product over a power

Expand the quotient, product, and denominator power in that order.

log⁡3(abc2)\log_3\left(\frac{ab}{c^2}\right)

Expected result

log⁡3a+log⁡3b−2log⁡3c\log_3a+\log_3b-2\log_3c

Sum stays grouped

There is no valid logarithm rule for splitting a sum.

ln⁡(x+4)\ln(x+4)

Expected result

ln⁡(x+4),x>−4\ln(x+4),\quad x>-4

Even power without a positivity assumption

Use absolute value to preserve the full real domain x not equal to zero.

ln⁡(x2)\ln(x^2)

Expected result

2ln⁡∣x∣,x≠02\ln|x|,\quad x\ne0

Complete example

Expand a nested quotient without losing a minus sign

Assume x is nonzero and y is greater than negative 1. The denominator is positive, and the numerator 4x squared is positive.

ln⁡(4x2y+1)\ln\left(\frac{4x^2}{\sqrt{y+1}}\right)
  1. 1
    Separate numerator and denominator

    The entire denominator logarithm receives a minus sign.

    ln⁡(4x2)−ln⁡((y+1)1/2)\ln(4x^2)-\ln\left((y+1)^{1/2}\right)
  2. 2
    Expand the numerator product

    Four and x squared are positive on the stated domain.

    ln⁡4+ln⁡(x2)−12ln⁡(y+1)\ln4+\ln(x^2)-\frac12\ln(y+1)
  3. 3
    Handle the even power safely

    Because x may be negative, use ln(x squared)=2ln|x|.

    ln⁡4+2ln⁡∣x∣−12ln⁡(y+1)\ln4+2\ln|x|-\frac12\ln(y+1)
ln⁡4+2ln⁡∣x∣−12ln⁡(y+1)\ln4+2\ln|x|-\frac12\ln(y+1)

Verification: Condensing produces ln(4|x|^2/(y+1)^(1/2)); since |x|^2=x^2, this equals the original expression on x not equal to zero and y>-1.

Avoidable errors

Common mistakes and how to fix them

Splitting addition inside a logarithm

Problem: Write log(x+2)=log(x)+log(2).

Why it matters: The product rule does not apply to a sum.

Better approach: Leave x+2 grouped and state x+2>0.

Dropping the quotient minus sign

Problem: Expand log(x/y) as log(x)+log(y).

Why it matters: Division corresponds to subtracting exponents, so it produces a difference of logarithms.

Better approach: Write log(x)-log(y) and carry the minus through any denominator powers.

Moving a power inside the argument

Problem: Rewrite log(x^3) as (log x)^3.

Why it matters: The power becomes a coefficient outside the logarithm, not an exponent on the logarithm value.

Better approach: Use log(x^3)=3log(x) under the required domain conditions.

Assuming every variable is positive

Problem: Write ln(x^2)=2ln(x) when x may be negative.

Why it matters: The original expression is defined for negative nonzero x, but ln(x) is not.

Better approach: Use 2ln|x| unless the problem explicitly states x>0.

Trust the result for the right reasons

Checks, assumptions, and limits

How results are checked

  • Each rewrite is limited to a product, quotient, or power identity.
  • The expanded result is condensed independently under the same assumptions.
  • Original and separated arguments are checked for positivity.
  • Even powers use absolute values when variable positivity is not given.

When to stop and revise the input

  • Sums and differences inside an argument are not distributed across logarithms.
  • Sign-unknown symbolic factors may require assumptions before complete expansion.
  • The calculator works with real logarithms, not complex branches.
  • Equivalent expansions can look different when constant factors are simplified first.

Common questions

Expand a logarithmic expression FAQ

What are the three rules used to expand logarithms?

A product becomes a sum, a quotient becomes a difference, and an exponent becomes a coefficient, provided all logarithms are defined on the stated domain.

Can I expand log(x+y)?

No general logarithm rule separates a sum. Keep x+y inside one logarithm unless the sum can first be factored into a product.

Why does an absolute value sometimes appear?

An expression such as ln(x squared) is defined for negative nonzero x, while ln(x) is not. Writing 2ln|x| preserves the original domain.

Does expansion change the value of the expression?

No. It rewrites the same value under the same base and domain conditions. Condensing the result should reconstruct the original expression.

Should I simplify constants before expanding?

Either order can be valid. Simplifying a constant power first may make the answer shorter, while expanding first can make the property used more visible.

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.