1. Let's state the problem clearly: We want to analyze expressions where two different bases are raised to the same exponent. For example, consider the expression $$a^x = b^x$$ where $$a$$ and $$b$$ are different bases and $$x$$ is the same exponent. 2. Assume $$a$$ and $$b$$ are positive real numbers and $$a \neq b$$. We want to understand under what conditions $$a^x = b^x$$ holds. 3. To solve for $$x$$, take the natural logarithm on both sides: $$\ln(a^x) = \ln(b^x)$$ Using logarithm properties: $$x \ln(a) = x \ln(b)$$ 4. Simplify this expression: $$x (\ln(a) - \ln(b)) = 0$$ 5. This equation implies either: - $$x = 0$$ or - $$\ln(a) = \ln(b)$$ (which means $$a = b$$). 6. Since the bases $$a$$ and $$b$$ are different (i.e., $$a \neq b$$), the only solution is: $$x = 0$$ 7. Conclusion: For two different bases raised to the same exponent to be equal, the exponent must be zero. In other words, $$a^x = b^x$$ implies $$x = 0$$ when $$a \neq b$$. Final Answer: $$x = 0$$ is the only solution if $$a \neq b$$.