Find the exact solution to the equation

Question

Find the exact solution to the equation below. Do not give a decimal approximation 

(log(x⁴)+log(x⁶))/(log(5x))=7

x= 

Answer 1

(log(x⁴)+log(x⁶))/(log(5x))=7

 

Identity: log(x^a) = alog(x)

 

[4log(x)+6log(x)]/log(5x) = 7

 

Identity: log(ab) = log(a)+log(b)

 

10log(x)/[log(5)+log(x)] = 7

 

10log(x) = 7log(5) + 7Log(x)    [Multiply both sides by log(5)+log(x)]

 

10log(x)-7log(x) = 7 log(5)

 

3log(x) = 7log(5)

 

log(x) = (7/3)log(5) = log(5^7/3)      [Identity: b*log(a) = log(a^b)]

 

x = 5^7/3