algebra II method of common bases?

You are given:

y = (1/25)^[(x - 2) / 5] - 10

And the point (a, 115) and need to find the value of a.

We know the value of y (115), so we can find the unknown (a):

115 = (1/25)^[(a - 2) / 5] - 10

Add 10 to both sides:

125 = (1/25)^[(a - 2) / 5]

125 and 1/25 are both powers of 5.  We can substitute those values:

5³ = (5⁻²)^[(a - 2) / 5]

The exponent of an exponent is the same as the product of the two exponents:

5³ = 5^{-2[(a - 2) / 5]}

Now that we have two values equal that has the same base, the exponents must also be the same:

3 = -2[(a - 2) / 5]

This should be simple now.  Multiply both sides by 5, then divide both sides by -2:

15 = -2(a - 2)
-15/2 = a - 2

Finally, add 2 to both sides and simplify:

2 - 15/2 = a
4/2 - 15/2 = a
-11/2 = a...Show more