SOLUTION: Suppose f and g are functions respectively defined by f(x) = x -2 and g(x) = x. Find all positive x for which 7^(g(x) ⋅ log_7 f(x)) = f(x)

Algebra ->  Functions -> SOLUTION: Suppose f and g are functions respectively defined by f(x) = x -2 and g(x) = x. Find all positive x for which 7^(g(x) ⋅ log_7 f(x)) = f(x)      Log On


   

Question 1186900: Suppose f and g are functions respectively defined by f(x) = x -2 and g(x) = x. Find all positive x for which 7^(g(x) ⋅ log_7 f(x)) = f(x)
Answer by greenestamps(13214) About Me  (Show Source):
You can put this solution on YOUR website!


7%5E%28g%28x%29log%287%2Cf%28x%29%29%29=f%28x%29

Substitute f(x)=x-2; g(x)=x

7%5E%28x%2Alog%287%2C%28x-2%29%29%29=x-2

Use a%5E%28bc%29=%28a%5Eb%29%5Ec to rewrite the left hand side:

%287%5Elog%287%2C%28x-2%29%29%29%5Ex=x-2

Use a%5E%28log%28a%2Cb%29%29=b to again rewrite the left hand side

%28x-2%29%5Ex=x-2

Take logs of both sides:

x%2Alog%28%28x-2%29%29=log%28%28x-2%29%29

x%2Alog%28%28x-2%29%29-log%28%28x-2%29%29=0

%28x-1%29log%28%28x-2%29%29=0

x-1=0 or log%28%28x-2%29%29=0

x=1 or x-2=1

x=1 or x=3

x=1 does not satisfy the original equation because it results in trying to take the logarithm of a negative number.

x=3 satisfies the equation.

ANSWER: x=3