Solving Exponential Equations from the Definition

Since the bases are the same, then I can equate the powers and solve:

Then my solution is:

Since 9 = 3², this is really asking me to solve:

By converting the 9 to a 3², I've converted the right-hand side of the equation to having the same base as the left-hand side. Since the bases are now the same, I can set the two powers equal to each other:

In this case, I have an exponential on one side of the "equals" and a number on the other. I can solve the equation if I can express the "27" as a power of 3. Since 27 = 3³, then I can convert and proceed with the solution:

If I'm not sure of my answer, or if I want to check it before I hand it in (on, say, a test), I can check it by plugging it back into the original exercise. The power on the left-hand side of the original equation would simplify as:

And 3³ = 27, which is the right-hand side of the original equation. Then my (confirmed) solution is:

This exercise works just like the previous one:

So my answer is:

This equation is similar to the previous two but is not quite the same, because 8 is not a power of 4. However, both 8 and 4 are powers of 2, so I can convert. The right-hand side is easy:

The left-hand side is a bit messier:

Now that I've re-stated both sides as powers on 2, I can solve:

Negative exponents can be used to indicate that the base belongs on the other side of the fraction line. Since 64 = 4³, then I can use negative exponents to convert the fraction to an exponential expression:

Using this, I can solve the equation:

In order to be able to solve this one, I first need to recall that square roots are the same as one-half powers, and convert the radical to exponential form. Then I can solve the equation:

Then my answer is:

Think about it: What power on the positive number "2" could possibly yield a negative number? A number can never go from positive to negative by taking powers; I can never turn a positive two into a negative anything, four or otherwise, by multiplying two by itself, regardless of the number of times I do the multiplication. Exponentiation simply doesn't work that way. So the answer here is: