Solve the exponential Equation.

Question

Solve the exponential Equation.

 

 

    x+4

5^      = 8^x

 

( Round to the nearest thousandth as needed. Use a comma to separate Answers as needed.)

Answer 1

 5 ^( X+4 ) = 8 ^ x

 

   Take log of both sides:

 

  ( X +4 ) log 5 = x log 8

 

 

      (X+ 4 ) / X = Log 8 / Log 5

 

 

      ( X + 4 ) / X = 1.292

 

        X +4 = 1.292 X

 

          X = 4/0.292 = 13.6986

               

 

        Test:

    

          5 ^ 17.6986 =2.3499752*10^12    

 

           8 ^ 13.6986 =2.35000428 * 10^12           

Answer 2

5^(x+4) = 8^x

 

Take the log of both sides:

 

log(5^(x+4)) = log(8^x)

 

Use the power rule:

 

(x+4) log(5) = x log(8)

 

x log(5) + 4 log(5) = x log(8)

 

4 log(5) = x log(8) - x log(5) = x log(8/5)

 

x = 4 log(5) / log(1.6)

 

x ≈ 13.69723816828372 ≈ 13.697

 

check:

 

5^((13.69723816828372)+4) =? 8^(13.69723816828372)

 

2343357986099.765 ≈ 2343357986100.124  √