Solving Logarithm Equations
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Solving logarithm equations are similar to exponential
equations. First, we isolate the
logarithm on one side by itself with a coefficient of one. Then we use Property 4 from the Logarithm Properties section with an appropriate
choice of b. In choosing the appropriate b, we need to remember that the b MUST match the base on the logarithm!
Solve each of the following equations.
1.
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The first step is to divide by the
4, then we’ll convert to an exponential equation.
Note that since we had a common
log in the original equation we were forced to use a base of 10 in the
exponential equation. Once we’ve used
Property 4 to simplify the equation we’ve got an equation that can be solved.
Now, with exponential equations we
were done at this point, but we’ve got a little more work to do in this
case. Recall the answer to the domain of
a logarithm (the answer to Problem 9 in the Logarithm
Properties section). We can’t take
the logarithm of a negative number or zero.
This does not mean that can’t be a solution just because it’s negative
number! The question we’ve got to ask is
this : does this solution produce a negative number (or zero) when we plug it
into the logarithms in the original equation.
In other words, is negative or zero if we plug into it?
Clearly, (I hope…) will be positive when we plug in.
Therefore the solution to this is .
Note that it is possible for logarithm
equations to have no solutions, so if that should happen don’t get to excited!
2.
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There’s a little more
simplification work to do initially this time, but it’s not too bad.
Now, solve this.
I’ll leave it to you to check that
will be positive upon plugging into it and so we’ve got the solution to the
equation.
3.
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This one is a little different
from the previous two. There are two
logarithms in the problem. All we need
to do is use Properties 5 7 from the
Logarithm Properties section to simplify
things into a single logarithm then we can proceed as we did in the previous
two problems.
The first step is to get
coefficients of one in front of both logs.
Now, use Property 6 from the Logarithm
Properties section to combine into the following log.
Finally, exponentiate both sides
and solve.
Finally, we just need to make sure
that the solution, ,
doesn’t produce negative numbers in both of the original logarithms. It doesn’t, so this is in fact our solution
to this problem.
4.
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This one is the same as the last
one except we’ll use Property 5 to do the simplification instead.
So, potential solutions
are and . Note, however that if we plug into either of the two original logarithms we
would get negative numbers so this can’t be a solution. We can however, use .
Therefore, the solution to this
equation is .
It is important to check your
potential solutions in the original
equation. If you check them in the
second logarithm above (after we’ve combined the two logs) both solutions will
appear to work! This is because in
combining the two logarithms we’ve actually changed the problem. In fact, it is this change that introduces
the extra solution that we couldn’t use!
So, be careful in finding
solutions to equations containing logarithms.
Also, do not get locked into the idea that you will get two potential
solutions and only one of these will work.
It is possible to have problems where both are solutions and where
neither are solutions.