Solve simple exponential equations with common bases

The understanding is that equations such as $2^x=8$ or $3^{2x-1}=27$ can be solved by expressing both sides with the same positive base, then using the one-to-one nature of exponential functions to equate their exponents and verify the resulting value. This includes recognizing equivalent forms involving integer exponents and reciprocals, but not equations requiring logarithms, unlike bases that cannot be readily rewritten, or more advanced multi-term exponential equations.

Detailed Explanation: Solve simple exponential equations with common bases

To solve an exponential equation with common bases:

Rewrite both sides using the same positive base.
Set the exponents equal because exponential functions with the same base are one-to-one.
Solve the resulting equation.
Check the answer in the original equation.

Example

Solve:

3^{2x-1}=27

Rewrite $27$ as a power of $3$ :

27=3^3

So the equation becomes:

3^{2x-1}=3^3

Since the bases are the same, set the exponents equal:

2x-1=3

Add $1$ to both sides:

2x=4

Divide by $2$ :

x=2

Check the answer:

3^{2(2)-1}=3^3=27

The solution is:

\boxed{x=2}

Learn by doing: Solve simple exponential equations with common bases

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Algebra with Exponents - Binomial and Monomial

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