Question

solve the equation $2^{(3x - 1)} \cdot 4^{(5x - 7)} = 8^{(x + 4)}$

Explanation:

Step1: Rewrite bases as powers of 2

$2^{(3x-1)} \cdot (2^2)^{(5x-7)} = (2^3)^{(x+4)}$

Step2: Simplify exponents of powers

$2^{(3x-1)} \cdot 2^{(10x-14)} = 2^{(3x+12)}$

Step3: Combine left side exponents

$2^{(3x-1+10x-14)} = 2^{(3x+12)}$
$2^{(13x-15)} = 2^{(3x+12)}$

Step4: Set exponents equal

$13x - 15 = 3x + 12$

Step5: Isolate x terms

$13x - 3x = 12 + 15$
$10x = 27$

Step6: Solve for x

$x = \frac{27}{10}$

Answer:

$x = \frac{27}{10}$ or $x=2.7$