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Solve exponential equations using common bases

The learner understands that exponential equations can be solved by rewriting both sides with a common positive base, then applying the one-to-one property: for 0<a≠10<a\ne1, au=ava^u=a^v implies u=vu=v. This reasoning supports solving equations with linear exponents and integer or rational powers while preserving equivalence; cases requiring logarithms, nonreal solutions, or more generalized exponential methods are not included.

Detailed Explanation: Solve exponential equations using common bases

To solve an exponential equation using common bases:

  1. Rewrite both sides using the same positive base.
  2. Use the one-to-one property: if (0<a≠1)(0<a\ne 1) and (au=av)(a^u=a^v), then (u=v)(u=v).
  3. Solve the resulting equation for the variable.
  4. Check the answer if needed.

Example: Solve

4x+1=8x−14^{x+1}=8^{x-1}

Rewrite 44 and 88 as powers of the common base 22:

4=22and8=234=2^2 \qquad\text{and}\qquad 8=2^3

Substitute these into the equation:

(22)x+1=(23)x−1(2^2)^{x+1}=(2^3)^{x-1}

Use the power-of-a-power rule, ((am)n=amn)((a^m)^n=a^{mn}):

22(x+1)=23(x−1)2^{2(x+1)}=2^{3(x-1)}

Since both sides have the same base 22, set the exponents equal:

2(x+1)=3(x−1)2(x+1)=3(x-1)

Solve:

2x+2=3x−32x+2=3x-3 2=x−32=x-3 x=5x=5

Therefore, the solution is

x=5\boxed{x=5}

Check:

45+1=46=40964^{5+1}=4^6=4096

and

85−1=84=40968^{5-1}=8^4=4096

Both sides are equal, so (x=5)(x=5) is correct.

Learn by doing: Solve exponential equations using common bases

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    Algebra with Exponents - Binomial and Monomial

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