SOLUTION: Write the given expression as a single logarithm. 4ln(x)-2(ln(x^3)+4ln(x))

Question 255834: Write the given expression as a single logarithm.
4ln(x)-2(ln(x^3)+4ln(x))
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

We have an expression of three logarithms and we want an expression of one logarithm. This means that we will need to eliminate and/or combine the logarithms until we are down to just one.

Logarithms can be eliminated (i.e. replaced with an equivalent, non-logarithmic expression) if it is a logarithm we can find "by hand". Examples of these are:

Logarithms can be combined using one of the following properties of logarithms:

These properties require that the two logarithms to be combined have coefficients of 1. For logarithms with other coefficients we have another property, , which allows us to move coefficients into the argument as an exponent.

Your expression has no logarithms we can eliminate. So we will have to use the properties exclusively to combine these into one. But before we start this, let's use the Distributive Property to simplify the original expression:

Now, as mentioned earlier, we have to get coefficients of 1 in front of the logs before we try to combine them. Using the third property to move the coefficients into the arguments as exponents we get:

Using a rule for exponents on the middle log we get:

Now our coefficients are 1's. Now we can start combining them. Until you get good at this, you should do this two logs at a time (instead of all three at once). Because it is a subtraction, we will use the second property listed above to combine the first two logarithms:

Using a rule for exponents this simplifies to:

Again because of the subtraction, we will use the second property again to combine the remaining two logs:

Using our rule for exponents again we get:

This may be an acceptable answer. It is a single logarithm after all. We could rewrite it in various forms:
Without a negative exponent:
Use the third property of logarithms above to move the coefficient out in front:
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