3^(2x+5)=4^(1-3x)

Question

Solve for x

Answer 1

Step 1: Take the natural log of both sides using the identity log(a^b) = b*log(a).

log(3)(2x+5)=2log(2)(1-3x)

 

Step 2: On the left hand side, expand the terms.

2log(3)x + 5log(3) = 2log(2)(1-3x)

 

Step 3: On the right hand side, expand the terms.

2log(3)x + 5log(3) = 2log(2) - 6log(2)x

 

Step 4: Isolate x to the left side.

(6log(2)+2log(3))x=2log(2)-5log(3)

 

Step 5: Solve for x. (Calculator is needed for decimal answer).

x = 2log(2) - 5log(3)

6log(2) + 2log(3)

 

x is approximately -0.64611

Answer 2

3^2x+5 = 4^1-3x

 

Take ln of both sides (you could also use base 10 logs, or any other base as well).  

 

ln(3^2x+5) = ln(4^1-3x)

 

(2x+5)ln3 = (1-3x)ln4

 

2xln3 + 5ln3 = ln4 - 3xln4

 

2xln3 + 3xln4 = ln4 - 5ln3

 

x(2ln3 + 3ln4) = ln4 - 5ln3

 

x = (ln4 - 5ln3)/(2ln3 + 3ln4)

 

   = (ln4 - ln243)/(ln9 + ln64) ≈ -0.646