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All About That Base (Exponential Equations)

Meghan-Trainor-608x383

Solving exponential equations with different bases can be a little trickier than solving exponential equations with the same base.  Let’s have a look at this equation:

3x=2x-1

Not only are the bases, 3 and 2, not the same, we can’t even rewrite them to be the same base, like we could of this was our equation:

4x=2x-1
22x=2x-1

So what’s a poor Brightstormer to do? Well I’ll tell ya. First, we have to get every base with an x in the exponent on one side of the equation, and everything else on the other side. We split apart the crazy stuff in the exponent of 2 using the rule

base

Like this:

base2

Now we can multiply both sides by 2, and get this equation:

base3

Then divide both sides by 3x:

base4

Whew.  That was a alot.  But we’re in really good shape now.  Remember this rule of exponents?

base5

Let’s apply it to the left hand side of our equation.  We get

base6

Suh-weet! Now we can take the natural log of both sides of the equation (or you can use log base 10 AKA the standard logarithm… it’s your choice!)

base7

Use the power rule for logarithms to pull the x out front…

base8

Divide both sides by ln(2/3) to get x all by itself…

base9

And we’re done! Ta-da!  You can put the left hand side of the above equation into your calculator if you need to get an actual number for some reason.  If you’re still having trouble with solving exponential equations with different bases, check out these Brightstorm videos

http://www.brightstorm.com/math/precalculus/exponential-and-logarithmic-functions/solving-exponential-equations-with-the-different-bases/

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