LEMMA

Exponentials & logs

Logarithm properties

ALEKS placement

Logs turn multiplication into addition and exponents into coefficients.

What this covers

log(xy)=logx+logy,logyx​=logx−logy,logxn=nlogx
  • Use one base greater than 0 and different from 1. These expansion rules assume x and y are positive, so every individual log is defined.
  • Expand by giving each factor its own log term.
  • Turn each exponent into a coefficient out front.
  • Condense by reversing: coefficients become exponents, sums become products.
  • Keep every log on the same base.

Worked example

Worked example

Expand log(x³y / z), assuming x, y, and z are positive.

  1. logx3+logy−logzA product becomes a sum and a quotient becomes a difference.
  2. 3logx+logy−logzThe exponent moves to the front.
Another worked example

For x > 0 and y > 0, expand log(3x²/√y).

  1. log3+log(x2)−log(y1/2)Positive factors permit the product and quotient laws. √y is y to the one-half power.
  2. log3+2logx−21​logyWith positive x and y, move the powers outside as multipliers.

A common mistake

A mistake Lemma catches

From

log2​(x−1)+log2​(x+1)=log2​(2x)log2​(x−1)+log2​(x+1)=log2​(x2−1)

“Adding logs multiplies their insides: log₂ a + log₂ b = log₂(ab), so the inside is (x − 1)(x + 1) = x² − 1. Adding the insides, (x − 1) + (x + 1) = 2x, uses a rule that does not exist. At x = 3: log₂2 + log₂4 = 1 + 2 = 3, but log₂6 ≈ 2.58.”

Try one

Sample problem

Expand using logarithm properties. Assume x > 0 and y > 0.

log3​(xy)

Practice log properties free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

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