Solve for x 169x 13x 79SolutionWell use the substitution m

Solve for x: 169^x + 13^x = 79

Solution

We\'ll use the substitution method to solve the exponential equation.

13^x=t

169^x=(13^2)^x=(13^x)^2=t^2

t^2 + t = 79

We\'ll subtract 79 both sides:

t^2 + t - 79 = 0

We\'ll apply the quadratic formula:

t1 = [-1+sqrt(1+316)]/2=16/2=approx. 8

13^x=t1

13^x=8

We\'ll take logarithms both sides:

log 13^x = log 8

We\'ll use the power property of logarithms:

x log 13 = log 8

We\'ll divide by log 13:

x = log 8/log 13

t2=(-1-17)/2=approx. -9

13^x>0 so it\'s impossible that 13^x=-9!

The only solution is x=log 8/log 13.