SOLUTION: Solve the equation 7^(x � 2) = 5^x. Round the answer to four decimal places. Thanks. :)

Question 560326: Solve the equation 7^(x � 2) = 5^x. Round the answer to four decimal places. Thanks. :)
Answer by AnlytcPhil(1806) (Show Source): You can put this solution on YOUR website!

7^{x - 2} = 5^x

Take logarithms of both sides:

log(7^{x - 2}) = log(5^x)

Use a rule of lagarithms that allows an exponent to come 
out in front as a multiplier

(x - 2)log(7) = x�log(5)

To make the algebra less bulky and cumbersome, 
let A = log(7) and B = log(5), then we have

(x - 2)A = x�B

A(x - 2) = Bx

Ax - 2A = Bx

Get both x terms on the left side 
and the non-x term on the right side:

Ax - Bx = 2A

Factor out x

x(A - B) = 2A

Divide both sides by (A - B)

 = 

 = 

x = 

Substitute log(7) for A and log(5) for B

x = 

x = 11.56654212 round to 11.5665

Edwin

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