LEMMA

Exponentials & logs

Logarithmic equations

ALEKS placement

Condense to a single log, convert to exponential form, then check the domain.

What this covers

  • Use log properties to reach one log on one side.
  • Rewrite log_b(u) = c as b^c = u.
  • Solve the resulting equation.
  • Reject any root that makes an original argument zero or negative.

Worked example

Worked example

Solve log₂(x) + log₂(x - 2) = 3.

  1. log2​(x(x−2))=3,x>2A sum of logs is the log of the product, but only where both original logs exist: x > 2.
  2. x(x−2)=8,x>2Convert to exponential form, since 2³ = 8.
  3. x=4x² - 2x - 8 = 0 gives 4 and -2, but -2 makes log₂(x) undefined.
Another worked example

Solve log₂(x + 8) = 4.

  1. x+8>0A logarithm's argument must be positive. Record that before solving.
  2. x+8=24=16log₂(A) = 4 means 2⁴ = A.
  3. x=8Subtract 8. Check: 8 + 8 = 16 is positive, and log₂(16) = 4.

A common mistake

A mistake Lemma catches

From log2​(x(x−2))=3

x(x−2)=9x(x−2)=8

“log₂(stuff) = 3 means 2³ = stuff: the base is raised to the answer. 2³ = 2 · 2 · 2 = 8, so x(x − 2) = 8. Writing 9 computed 3² instead.”

Try one

Sample problem

Solve for x.

log3​(x)=5

Practice log equations 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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