Solve logarithms

Question

ln x + ln (2x) = log₅ 1

Answer 1

Use the following properties of logarithms 

 

ln(A) + ln(B) = ln(AB) 

log_b1 = 0 

 

to get ln(2x^2) = 0  

 

Now exponentiate both sides to get 

 

e^{ln(2x^2)} = e⁰

2x^2 = 1

x^2 = 1/2

x = +- sqrt(1/2) 

 

The negative solution is extraneous since you cannot take the ln of a negative number so 

 

x = sqrt(1/2) = 1/sqrt(2) = sqrt(2)/2 

 

Check:  ln(sqrt(2)/2) = -0.34657359

            ln(sqrt(2)) = 0.34657359