Solving Exponential Equations
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In each of the equations in this section the problem is
there is a variable in the exponent. In
order to solve these we will need to get the variable out of the exponent. This means using Property 3 and/or 7 from the Logarithm
Properties section above. In most
cases it will be easier to use Property 3 if possible. So, pick an appropriate logarithm and
take the log of both sides, then use Property 3 (or Property 7) where appropriate to simplify. Note that often some simplification will need
to be done before taking the logs.
Solve each of the following equations.
1.
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2.
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Now, in this case it looks like
the best logarithm to use is the common logarithm since left hand side has a
base of 10. There’s no initial
simplification to do, so just take the log of both sides and simplify.
At this point, we’ve just got a
quadratic that can be solved
So, it looks like the solutions in
this case are and .
As with the last one you could use
a different log here, but it would have made the quadratic significantly
messier to solve.
3.
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There’s a little more initial
simplification to do here, but other than that it’s similar to the first
problem in this section.
Now, take the log and solve. Again, we’ll use the natural logarithm here.
4.
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This one is a little different
from the previous problems in this section since it’s got x’s both in the exponent and out of the exponent. The first step is to factor an x out of both terms.
DO NOT DIVIDE AN x FROM BOTH TERMS!!!!
So, it’s now a little easier to
deal with. From this we can see that we
get one of two possibilities.
The first possibility has nothing
more to do, except notice that if we had divided both sides by an x we would have missed this one so be
careful. In the second possibility we’ve
got a little more to do. This is an
equation similar to the first few that we did in this section.
Don’t forget that !
So, the two solutions are and .