Solve the following exponential equation 25x 5X1 66 0 What

Solve the following exponential equation. 25^x + 5^X+1 -66 = 0 What is the exact solution? Select the correct choice below and fill in any answer choice X =(Type an exact answer using radicals and log functions as needed.) There is no solution What is the decimal approximation any answer boxes in your choice. for the solution? Select the correct choice below a x (Round to three decimal places as needed)

Solution

25^x + 5^(x+1) -66 = 0

=> (5²)^x + 5^(x+1) - 66 = 0

=> (5^x)² + 5^x. 5 - 66 = 0

Let 5^x = y

Then y² + 5y - 66 = 0

=> y² + 11y - 6y - 66 = 0

=> y(y+11) -6(y+11) = 0

=> (y-6)(y+11) = 0

=> y=6, y=-11

=> 5^x = 6 and 5^x = -11 (not possible)

Hence 5^x = 6 => xln(5) = ln(6)

=> x = ln(6)/ln(5) is exact solution

=> x = 1.11 is decimal approximation