Solving Exponential Equations
It's time to get properly equipped so you can face off with Expo. Thankfully, exponential equations have a few weak spots that you can take advantage of in your epic battles to come.
Like we mentioned before, the base is the source of the exponential's power, so we'll focus on that. Here's the first lesson: Strike at the base.
For b > 0, if bx = by, then x = y.
Yep, we can drop the base from both sides if they match. The reason this handy trick works, by the way, is because we can take the logarithm of both sides, which cancels out the bases.
bx = by
logb(bx) = logb(by)
x = y
For an equation like 4y = 4(x + 5), you could simply throw the 4s out the window to get y = x + 5. (Make sure they don't hit anything on the way down. The neighbors won't be happy.)
If the bases aren't the same, but share a common factor, you can just change the bases so they match. Just don't forget about your exponent properties. For example:
27(4x + 1) = 92x
Both 27 and 9 have 3 as a factor, so let's change them both to powers of 3.
(33)(4x + 1) = (32)2x
3(12x + 3) = 34x
Now that we've got matching bases on both sides, we can drop 'em and just keep the exponents.
12x + 3 = 4x
8x = -3
x = -3/8
Give it a shot: try solving for x with this equation, where the bases don't equal each other:
27x = 8(2x + 7)
Start by turning that 8 into 23.
27x = 23(2x + 7)
27x = 2(6x + 21)
7x = 6x + 21
x = 21