SOLUTION: How do I rewrite this expression as a logarithm of a single quantity? 3 (Im x - 2 In (x^3 + 2))+ 4 In 5 = ??? And... How do I solve these equations? Thanks! 1.) 2 In x =

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Question 627466: How do I rewrite this expression as a logarithm of a single quantity?
3 (Im x - 2 In (x^3 + 2))+ 4 In 5 = ???
And...
How do I solve these equations? Thanks!
1.) 2 In x = 14 ??? (demonstrate how?)
2. log^2 x + log^2 (x + 2) = log^2 (x + 6)= (please show steps!)
FYI: The "2" represents the base of the log for question #2. Again, thank you.
- Cassidy
Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!

First of all, these are not "eye-en's", they are "ell-en's". "ell" for logarithm and "en" for natural. These are called natural logarithms.

There are two ways to combine logarithmic terms:
Addition or subtraction if they are like terms. (Like logarithmic terms have the same bases and same arguments.)
Use one of the following properties:
These properties require that the bases be the same and the coefficients are 1's.
Your logs all have the same base, e. But their arguments are different so we cannot use addition or subtraction to combine the terms.

The coefficients are not 1's either. So it seems that we could not use the properties to combine the terms either. But fortunately there is another property of logarithms, , which allows us to "move" a coefficient into the argument as its exponent. So we can use this property to make the terms suitable for the other two properties:

Since this simplifies to:

We can now use the second property to combine the logs in the parentheses. (We use the second property because it has a "minus" between the terms, just like our logs.) Using the second property we get:

Before we try to combine the remaining logs we must first use the third property to move the 3 out of the way:

Now we can use the first property (because of the "plus" between the terms) to combine the remaining terms:

or: