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Logarithm properties

Condense an expression into one logarithm

Enter logarithms with a common base. Coefficients become exponents, sums become products, and subtractions become quotients in one final argument.

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Try an example

Result

Move the coefficient to an exponent, then combine the sum and difference

log⁡(x2yz)\log\left(\frac{x^2y}{z}\right)
Conditions
  • The original terms require x>0, y>0, and z>0.
  • Every logarithm uses the selected base 10.
  • The selected base is positive and is not 1.

Steps

  1. Move the coefficient inside The power rule used in reverse turns 2 log base 10 of x into log base 10 of x squared.log⁡(x2)+log⁡y−log⁡z\log(x^2)+\log y-\log z
  2. Combine the added terms A sum of same-base logarithms becomes the logarithm of a product.log⁡(x2y)−log⁡z\log(x^2y)-\log z
  3. Combine the difference Subtraction becomes division inside the final argument.log⁡(x2yz)\log\left(\frac{x^2y}{z}\right)
Independent check

Expanding the final logarithm under x>0, y>0, and z>0 returns exactly 2 log base 10 of x plus log base 10 of y minus log base 10 of z.

Scope

What this logarithm condensation covers

To condense logarithms, first move each coefficient into its argument as a power. Combine added logarithms by multiplication and subtracted logarithms by division, without changing the base or dropping original domain conditions.

Combine logarithmic sums

Turn added logarithms with one base into the logarithm of a product.

Examples: log_b x+log_b y, ln 2+ln x

Combine logarithmic differences

Turn subtraction into a quotient while preserving all original positive-argument restrictions.

Examples: log_b x-log_b y, ln(x+1)-ln3

Move coefficients to powers

Use the power rule in reverse, including negative and fractional coefficients.

Examples: 3ln x, (1/2)log_b y, -2ln z

Identify expressions that cannot combine

Keep terms separate when their bases differ or when a nonlogarithmic term cannot be rewritten in the shared base.

Examples: log_2 x+log_3 y, ln x+5

How to use it

Enter enough information for one clear task

  1. 1
    Confirm one common base

    Every logarithm being combined must use the same base. A change-of-base expression is not automatically a same-base sum.

  2. 2
    Move coefficients first

    Rewrite c log_b(M) as log_b(M^c) before combining addition and subtraction.

  3. 3
    Translate signs into operations

    Use multiplication for plus signs and division for minus signs, preserving the original grouping.

  4. 4
    Expand to verify

    Apply the logarithm properties in the opposite direction and confirm that every coefficient and sign returns.

Worked inputs

Examples to try

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

Sum of two logs

Multiply the positive arguments.

log⁡5x+log⁡5y\log_5x+\log_5y

Expected result

log⁡5(xy)\log_5(xy)

Difference of two logs

Place the subtracted argument in the denominator.

ln⁡x−ln⁡4\ln x-\ln4

Expected result

ln⁡(x4),x>0\ln\left(\frac{x}{4}\right),\quad x>0

Coefficient becomes a power

Use the power rule in reverse.

3log⁡2x3\log_2x

Expected result

log⁡2(x3),x>0\log_2(x^3),\quad x>0

Fractional coefficient

Move one-half to a power and write the positive square root.

12ln⁡y\frac12\ln y

Expected result

ln⁡y,y>0\ln\sqrt y,\quad y>0

Multiple additions and a subtraction

Convert 2 to a power, multiply added arguments, and divide by the subtracted one.

log⁡7a+2log⁡7b−log⁡7c\log_7a+2\log_7b-\log_7c

Expected result

log⁡7(ab2c)\log_7\left(\frac{ab^2}{c}\right)

Negative coefficient

The negative powered term belongs in the denominator.

ln⁡x−2ln⁡y\ln x-2\ln y

Expected result

ln⁡(xy2)\ln\left(\frac{x}{y^2}\right)

Different bases stay separate

The product rule requires the same base.

log⁡2x+log⁡3y\log_2x+\log_3y

Expected result

cannot be condensed as one elementary logarithm\text{cannot be condensed as one elementary logarithm}

Complete example

Condense coefficients and grouped subtraction carefully

The original expression requires x>1 and y>0. Those stronger conditions stay with the condensed form.

12ln⁡x+ln⁡(x−1)−3ln⁡y\frac12\ln x+\ln(x-1)-3\ln y
  1. 1
    Move coefficients to exponents

    One-half produces a square root and 3 produces a cube.

    ln⁡(x1/2)+ln⁡(x−1)−ln⁡(y3)\ln(x^{1/2})+\ln(x-1)-\ln(y^3)
  2. 2
    Combine the addition

    Multiply the first two positive arguments.

    ln⁡(x(x−1))−ln⁡(y3)\ln\left(\sqrt{x}(x-1)\right)-\ln(y^3)
  3. 3
    Combine the subtraction

    Place y cubed in the denominator.

    ln⁡(x(x−1)y3)\ln\left(\frac{\sqrt{x}(x-1)}{y^3}\right)
ln⁡(x(x−1)y3),x>1, y>0\ln\left(\frac{\sqrt{x}(x-1)}{y^3}\right),\quad x>1,\ y>0

Verification: Expanding the quotient, product, square root, and cube returns one-half ln(x) plus ln(x-1) minus 3ln(y) on the original domain.

Avoidable errors

Common mistakes and how to fix them

Multiplying logarithm values instead of arguments

Problem: Turn log(x)+log(y) into log(x)log(y).

Why it matters: The sum becomes one logarithm whose argument is xy.

Better approach: Write log(xy), not a product of two logarithm values.

Putting a coefficient in front of the argument

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

Why it matters: A coefficient on a logarithm becomes an exponent on its argument.

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

Combining unlike bases

Problem: Turn log base 2 of x plus log base 3 of y into log of xy.

Why it matters: The product rule applies only when the logarithm base is the same.

Better approach: Keep the terms separate or change both to a common base before doing further algebra.

Weakening the original domain

Problem: Keep only the condition that the final quotient is positive.

Why it matters: Each original logarithm required its own argument to be positive before condensation.

Better approach: Carry every original positive-argument condition into the final answer.

Trust the result for the right reasons

Checks, assumptions, and limits

How results are checked

  • Only same-base logarithms are combined.
  • Every coefficient is converted to the matching argument exponent.
  • The condensed expression is expanded independently to recover all original terms.
  • Original positive-argument conditions are preserved even when the final argument suggests a wider domain.

When to stop and revise the input

  • Different logarithm bases are not silently treated as equal.
  • Nonlogarithmic terms stay separate unless an exact same-base logarithmic form is supplied.
  • Real logarithm rules are used; complex branch behavior is not included.
  • An expression with ambiguous grouping must be rewritten with parentheses.

Common questions

Condense an expression into one logarithm FAQ

What does it mean to condense logarithms?

It means rewriting a sum or difference of compatible logarithms as one logarithm without changing its value on the original domain.

Why must the bases match?

The product, quotient, and power properties describe one exponential base. Different bases represent different inverse functions and cannot use those rules directly.

Where does a subtracted logarithm go?

Its powered argument goes in the denominator of the final logarithm. Multiple subtracted terms multiply together in that denominator.

Why do the original domain conditions still matter?

Condensation rewrites an existing expression. It cannot retroactively make an original logarithm with a nonpositive argument valid.

How can I check a condensed answer?

Expand it using the quotient, product, and power rules. You should recover every original logarithm, sign, and coefficient under the same conditions.

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.