Question

Solve the following exponential equation Exact answer only.9^x+3^x+1-40=0

Answer 1

9^x+3^x+1-40=0=3^2x+3^x-39=0.

Let y=3^x, then y²+y-39=0. Using the quadratic formula:

y=(-1±√(1+156))/2=(-1±√157)/2=5.764982 approx. The other root is negative and can be ignored because 3^x>0.

3^x=5.764982, xlog(3)=log(5.764982), x=log(5.764982)/log(3)=1.594559 approx.

The exact answer would be x=log(½(√157-1))/log(3).