Solve Exponential Equations - Same Base Method - Algebra-1

1. Fundamental Concepts

Core Principle: If two exponential expressions with the same base are equal, their exponents must be equal. That is: If $a^{\text{something}} = a^{\text{whatever}}$ (where a > 0 and $a \neq 1$), then $\text{something} = \text{whatever}$. This is because the exponential function $y = a^x$ is injective (one-to-one), meaning each input (exponent) corresponds to a unique output (power value).

Looking for a Common Base: The core of solving such equations is to rewrite both sides of the equation with the same base. This can be done using properties of exponents, such as $a^m \cdot a^n = a^{m+n}$, $(a^m)^n = a^{mn}$, and $a^{-n} = \frac{1}{a^n}$.
Limitations of Applicability: This method only works when both sides of the equation can be rewritten with the same base. If this is not possible (e.g., $8^{4x+3} = 9^{2x}$, where 8 and 9 cannot be converted to a common base), the method is ineffective.

Problem: Solve $$2^x = 2^4$$

1: Solve $16^x = \left(\frac{1}{64}\right)$
Solution: Rewrite bases as powers of 2: $16 = 2^4$ and $\frac{1}{64} = 2^{-6}$. The equation becomes $2^{4x} = 2^{-6}$. Equating exponents: $4x = -6$, so $x = -\frac{3}{2}$.
2: Solve $6^{2x-1} = 216^{x+1}$
Solution: Note that $216 = 6^3$. Rewrite the equation as $6^{2x-1} = 6^{3(x+1)}$. Equating exponents: $2x - 1 = 3x + 3$, so $x = -4$.
3: Solve $\left(\frac{1}{64}\right) = \left(\frac{1}{8}\right)^{2x-5}$
Solution: (base $\frac{1}{8}$): $\frac{1}{64} = \left(\frac{1}{8}\right)^2$, so the equation becomes $\left(\frac{1}{8}\right)^2 = \left(\frac{1}{8}\right)^{2x-5}$. Equating exponents: $2 = 2x - 5$, so $x = \frac{7}{2}$.

Rewrite Bases: Observe the bases on both sides and use exponent properties (e.g., negative exponents, powers) to convert them to the same base (e.g., convert 16 and 64 to base 2, or 216 to base 6).
Equate Exponents: Once the bases are the same, set the exponents equal to each other to form a linear equation.
Solve and Verify: Solve the linear equation and substitute the solution back into the original equation to check for correctness (pay special attention to calculations involving negative or fractional exponents).
Check Applicability: If the bases cannot be converted to the same form (e.g., 8 and 9), avoid forcing this method and use alternatives like logarithms instead.