what is the value of x if (9^{x - 1}-2 = 25)?\n(x=\frac{1}{2})\n(x = 2)\n(x=\frac{5}{2})\n(x = 4)

Explanation:

Step1: Isolate the exponential term

Add 2 to both sides of the equation $9^{x - 1}-2 = 25$. $9^{x - 1}=25 + 2=27$

Step2: Rewrite the bases

Since $9 = 3^2$ and $27=3^3$, the equation $9^{x - 1}=27$ can be rewritten as $(3^2)^{x - 1}=3^3$. Using the power - of - a - power rule $(a^m)^n=a^{mn}$, we get $3^{2(x - 1)}=3^3$.

Step3: Set the exponents equal

Since the bases are the same, we can set the exponents equal to each other: $2(x - 1)=3$

Step4: Solve for x

Expand the left - hand side: $2x-2 = 3$. Add 2 to both sides: $2x=3 + 2=5$. Divide both sides by 2: $x=\frac{5}{2}$

Answer:

$x=\frac{5}{2}$