solve for x: (exact answers)

5^(2x+1)-26(5^x)+5=0

To solve the equation 5^(2x+1) - 26*(5^x) + 5 = 0 for x, we can use a substitution technique. Let's substitute y = 5^x to simplify the equation:

Now we have y^2 * 5 - 26y + 5 = 0. This is a quadratic equation in terms of y.

To solve quadratic equations, we can use the quadratic formula:

y = ( -b ± √(b^2 - 4ac) ) / 2a

In this case, a = 5, b = -26, and c = 5. Let's plug in these values:

y = ( -(-26) ± √((-26)^2 - 4 * 5 * 5) ) / (2 * 5)
y = (26 ± √(676 - 100)) / 10
y = (26 ± √(576)) / 10
y = (26 ± 24) / 10

Now we have two possible solutions for y:

1. y = (26 + 24) / 10 = 50 / 10 = 5
2. y = (26 - 24) / 10 = 2 / 10 = 0.2

Now we can substitute back the value of y into the equation y = 5^x:

1. 5 = 5^x
Taking the logarithm of both sides:
log(5) = x * log(5)
x = log(5) / log(5) = 1

2. 0.2 = 5^x
Taking the logarithm of both sides:
log(0.2) = x * log(5)
x = log(0.2) / log(5) ≈ -0.69897 / 0.69897 ≈ -1 (approximately)

So, the exact solutions for x are x = 1 and x ≈ -1.