Question

243^{x - 4} = 81^{x - 5}

Explanation:

Step1: Rewrite bases as powers of 3

$243 = 3^5$, $81 = 3^4$, so substitute:
$$(3^5)^{x-4} = (3^4)^{x-5}$$

Step2: Simplify exponents via power rule

Use $(a^m)^n = a^{mn}$:
$$3^{5(x-4)} = 3^{4(x-5)}$$

Step3: Set exponents equal (same base)

Since $3^a=3^b$ implies $a=b$:
$$5(x-4) = 4(x-5)$$

Step4: Expand both sides

$$5x - 20 = 4x - 20$$

Step5: Solve for x

Subtract $4x$ and add 20 to both sides:
$$5x - 4x = -20 + 20$$
$$x = 0$$

Answer:

$x = 0$