# Logarithms and Exponential Functions

# 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 *b ^{x}* =

*b*, then

^{y}*x*=

*y.*

For an equation like 4^{y} = 4^{x + 5}, you could simply throw the 4's 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 denominator, you can just change the bases like so:

27^{4x + 1} = 9^{2x}

(3^{3})^{4x + 1} = (3^{2})^{2x}

3^{12x + 3} = 3^{4x}

12*x* + 3 = 4*x*

8*x* = -3

*x* = -3/8

### Sample Problem

Give it a shot, try solving for *x *with this one where the bases don't equal each other:

2^{7x} = 8^{2x + 7}

2^{7x} = 2^{3(2x + 7)}

2^{7x} = 2^{6x + 21}

log_{2} 2^{7x} = log_{2} 2^{6x + 21}

7*x* = 6*x* + 21

*x* = 21