How can I solve a simple exponential equation by rewriting both sides with the same base?

It means expressing each side as a power of one shared base. For example, 4 and 8 can both be written using base 2: 4 = 2² and 8 = 2³. Then the exponent rules let you rewrite the whole equation using powers of 2.

The key condition is that the shared base is positive and not 1. For such a base, equal powers have equal exponents: if b^S = b^T, then S = T. [S1] This is the step that turns an exponential equation into an equation you can solve with ordinary algebra. If you cannot express both sides using a common base, this particular method may not apply directly.

Consider this equation:

4^(x+1) = 8^(x−1)

First, rewrite each number as a power of 2: 4 = 2² and 8 = 2³. Apply the power-of-a-power rule, (a^m)^n = a^(mn):

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

Now both sides have the same positive base, 2, which is not 1. Set the exponents equal:

2x + 2 = 3x − 3

Subtract 2x from each side, then add 3 to each side:

2 = x − 3 5 = x

So the solution is x = 5. The important detail is to multiply the outside exponent by the entire exponent in parentheses: 2(x+1) becomes 2x+2, not 2x+1.

Check your answer in the original equation, not only in the rewritten one. Put x = 5 into 4^(x+1) = 8^(x−1):

Left side: 4^(5+1) = 4^6 = 4096 Right side: 8^(5−1) = 8^4 = 4096

Both sides match, so 5 satisfies the original equation. You can also check the middle step: when x = 5, the exponents 2x+2 and 3x−3 are both 12, so both rewritten sides equal 2^12. The substitution check confirms the proposed solution for this example; the method alone does not establish that every equation can be rewritten with a shared base.

Use this sequence:

1. Look for a shared base that expresses both sides exactly.
2. Rewrite each side and simplify powers carefully, keeping parentheses intact.
3. Once the bases match and meet the condition above, set the exponents equal.
4. Solve that ordinary equation.
5. Substitute the answer into the original equation and compare the two sides.

For your written work, this compact format is useful:

Original equation: [equation] Common base: [base] Rewritten equation: [same-base form] Exponents equal: [equation in x] Solution: x = [value] Original-equation check: left side = [value]; right side = [value]

If a number is not an exact power of your chosen base, pause before forcing a rewrite. Choose a different shared base if one works; otherwise, this same-base routine may not be enough.