Solve: (sqrt 5)^x +1=25^x

Question

If you could please give detail on how to slove. I am having trouble learning this type of problem. Thank you!

Answer 1

The idea for solving this exponential equation (without a calculator) is to see if the powers can be written with the same base. If the powers are equal, then the exponents must be equal.

Since √5 is one of the bases, can 25^x be transformed to have the same base?

Since 25 = 5² and 5 = (√5)², then 25 =( (√5)²)²= (√5)⁴

So the exponential equation can be written as:

(√5)^x+1 = (√5)^4x

Since there are now equal powers with the same base, the exponents must be equal.

x+1 = 4x

3x = 1

x=1/3

There was some confusion on this problem because the presentation was interpreted as:

(√5)^x + 1 = 25^x

That equation has a solution, but it cannot be solved algebraically.

A technique learned in calculus called Newton's Method reveals that the answer to the latter interpretation is approximately:

x = 0.24786365066

Answer 2

The left hand side is equal to 5^{x/2}. The right hand side is equal to 5^{2x}. So we must have that x/2=2x or equivalently (3/2)x=0 or x=0.