for what value of x does 3^{2x}=9^{3x - 4}?\n1\n2\n3\n4

Explanation:

Step1: Rewrite 9 as 3^2

Since (9 = 3^2), then (9^{3x - 4}=(3^2)^{3x - 4}). By the power - of - a - power rule ((a^m)^n=a^{mn}), we have ((3^2)^{3x - 4}=3^{2(3x - 4)}=3^{6x-8}). The original equation (3^{2x}=9^{3x - 4}) becomes (3^{2x}=3^{6x - 8}).

Step2: Set the exponents equal

If (a^m=a^n), then (m = n) for (a>0,a\neq1). So, for (a = 3), we set (2x=6x - 8).

Step3: Solve for x

Subtract (2x) from both sides: (0 = 6x-2x - 8), which simplifies to (0 = 4x-8). Add 8 to both sides: (4x=8). Divide both sides by 4: (x = 2).

Answer:

2