LEMMA

Exponentials & logs

Evaluate logarithms

ALEKS placement

A logarithm asks one question: what exponent turns the base into this number?

What this covers

logb​x=y⟺by=x
  • Rewrite the log statement as an exponential one.
  • Look for the power of the base that matches.
  • The log of 1 is always 0, and the log of the base itself is 1.
  • Use change of base for anything a calculator has to handle.

Worked example

Worked example

Evaluate log₄(64).

  1. 4y=64Convert to exponential form.
  2. y=34³ = 64, so the exponent is 3.
Another worked example

Evaluate log₄(1/8) exactly.

  1. 4y=81​Name the unknown exponent y instead of treating log as division.
  2. 22y=2−3Use 4 = 2² and 1/8 = 2⁻³.
  3. 2y=−3⇒y=−23​Equate exponents of the same valid base.

A common mistake

A mistake Lemma catches

From 4y=64

y=16y=3

“A logarithm asks for the exponent: 4 to what power is 64? 4³ = 4 · 4 · 4 = 64, so y = 3. 64 ÷ 4 = 16 divides instead of finding a power — and 4¹⁶ is far more than 64.”

Try one

Sample problem

Evaluate exactly.

log3​(27)

Practice evaluate logs 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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