How To Solve Problems With Exponents

Recall the following logarithm property from the last section.\[ = ra\] Note that to avoid confusion with \(x\)’s we replaced the \(x\) in this property with an \(a\).\[ = \] We now have the same base and a single exponent on each base so we can use the property and set the exponents equal.

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One method is fairly simple but requires a very special form of the exponential equation.

The other will work on more complicated exponential equations but can be a little messy at times. This method will use the following fact about exponential functions. In this first part we have the same base on both exponentials so there really isn’t much to do other than to set the two exponents equal to each other and solve for \(x\).

So, the method we used in the first set of examples won’t work.

The problem here is that the \(x\) is in the exponent.

Now, in this case we don’t have the same base so we can’t just set exponents equal. \[ = \] Now, we still can’t just set exponents equal since the right side now has two exponents.

However, with a little manipulation of the right side we can get the same base on both exponents. If we recall our exponent properties we can fix this however.Because of that all our knowledge about solving equations won’t do us any good.We need a way to get the \(x\) out of the exponent and luckily for us we have a way to do that.That is not the problem that it might appear to be however, so for a second let’s ignore that.The real issue here is that we can’t write 8 as a power of 4 and we can’t write 4 as a power of 8 as we did in the previous part.The first thing to do in this problem is to get the same base on both sides and to so that we’ll have to note that we can write both 4 and 8 as a power of 2. \[\begin & = \frac\ & = \frac\end\] It’s now time to take care of the fraction on the right side.To do this we simply need to remember the following exponent property.If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.Now that we’ve seen the definitions of exponential and logarithm functions we need to start thinking about how to solve equations involving them.\[\frac = \] Using this gives, \[ = \] So, we now have the same base and each base has a single exponent on it so we can set the exponents equal.\[\begin2\left( \right) & = - 3\left( \right)\ 10 - 18x & = - 3x 6\ 4 & = 15x\ x & = \frac\end\] And there is the answer to this part.


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