LEMMA

Exponentials & logs

Exponential equations

ALEKS placement

Match the bases, then set the exponents equal.

What this covers

bu=bv⇒u=v
  • Rewrite both sides as powers of the same base if possible.
  • Set the exponents equal and solve.
  • Check whether one side is a power of the other's base: 27 = 3³, 16 = 2⁴, 125 = 5³.
  • Every equation in this topic can be written with matching bases; the tool for bases that never match comes two topics from now.

Worked example

Worked example

Solve 9^(x+1) = 27^x.

  1. 32(x+1)=33x9 = 3² and 27 = 3³.
  2. 2x+2=3xEqual bases mean equal exponents.
  3. x=2Solve the linear equation.
Another worked example

Solve 4^(x−1) = 8 over the reals.

  1. (22)x−1=23Rewrite both positive sides using base 2.
  2. 2x−2=3For a positive base other than one, equal powers have equal exponents.
  3. x=25​Add 2, then divide by 2. Check: 4^(3/2) = 8.

A common mistake

A mistake Lemma catches

From (22)x−1=23

x−1=32x−2=3

“The bases match only once 4 is written as 2²: (2²)^(x − 1) = 2^(2(x − 1)) = 2^(2x − 2). Then the exponents can be set equal: 2x − 2 = 3. x − 1 is the exponent on 4, not on 2.”

Try one

Sample problem

Solve for x.

3x=243

Practice exponential 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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