Solve the equation: 5^x=2*5^(x/2)+15

Question

I've tried using log but in vain.

Answer 1

5^x = 2(5^x/2) + 15

 

Since 5^x = (5^x/2)², we have (5^x/2)² - 2(5^x/2) - 15 = 0

 

    Let u = 5^x/2.  Then, u² - 2u - 15 = 0

 

                                 (u-5)(u+3) = 0

 

                                 u = 5 or u = -3

 

    That is, 5^x/2 = 5 or 5^x/2 = -3

 

    5^x/2 is always positive, so 5^x/2 can't be -3.

 

    Therefore, 5^x/2 = 5

 

                    5^x/2 = 5¹

 

                      x/2 = 1

 

                         x = 2