Solve the exponential equation. Express the solution set in terms of natural logarithms.

Question

4^x + 4 = 5^2x + 5

Answer 1

 

Takes the log of both sides.

By property of logs,

 

(x+4)(ln 4) = (2x+5) ( ln 5)

 

x * ln 4 +  4*ln 4 = (2x) * ln 5  + 5*ln 5

 

x * ln 4 - (2x) * ln 5  = 5 ln 5 - 4 ln 4

 

x [ ln 4 - 2 ln 5] = 5 ln 5 - 4 ln 4]

 

x = [ 5 ln 5 - 4 ln 4] / [ ln 4 - 2 ln 5]

 

   = ln (3125/16) / ln (4/25)

 

   or approximately -1.3653