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The key to the solution is to recognize that 4 and 8 are both powers of 2. So we can rewrite each of these logarithms in terms of base 2 logarithms. You might be able to realize that since and and since logarithms are exponents, that and . If this is hard to understand then here is some Algebra to show it. We will use a temporary variable:
Let
Rewriting this in exponential form we get:
Replace 4 with :
Use the property of exponents, :
Find the base 2 logarithm of each side:
Using a property of logarithms, , we can move the exponent out front:
Since by definition:
Multiply both sides by 1/2:
Replace our temporary variable with what it represents:
Similar logic shows that
So we can write our equation using base 2 logarithms:
Now we can use the property of logarithms used earlier, in the other direction, to move the coefficients back into the arguments as exponents:
Now that we have two base 2 logarithms that are equal, their arguments must be equal:
To solve this we'll raise both sides to the 6th power. (You'll see why in a minute.)
which simplifies to
(See why we used 6 now?)
Now we simplify
and solve. With a cubed term, the way to solve this is to get one side equal to zero and factor. Subtract from each side:
Factor out the GCF (which is ):
Now we can use the Zero Product Property which says that this product can be zero only if one of the factors is zero. So: or
Solving each of these we get or
With logarithmic equations we should always check our answers. We must make sure that the solutions do not make an argument to any logarithm zero or negative.
Checking x = 0:
As we can see, when x = 0 we get arguments that are zero so we have to reject x = 0 as a solution.
Checking x = 16:
which simplifies to
which simplifies to Check!
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