SOLUTION: 5 log 3 + log 4 in single logarithm

Question 815816: 5 log 3 + log 4 in single logarithm

Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

To go from two logs to one we need to either eliminate one of them or find a way to combine them. There's no way to eliminate a log from this expression so we will have to combine them.
There are two ways to combine logs:
- Algebraically add them. This requires that the bases and arguments of the logs are the same. The bases of our logs are the same, 10, but the arguments are different, 3 and 4. Since there is no way to make the arguments be the same we will not be able to add the logs algebraically.
- Use the property of logs. This property requires that the logs have the same bases (which ours do) and have coefficients of 1 (which ours do not since the first log has a coefficient of 5). Fortunately there is a way to change the coefficient to a 1.
A property of logs, , allows one to "move" a coefficient into the argument as its exponent. This is how we will handle the 5.

So our plan is to move the 5 using one property and then combine the logs using the other:

which simplifies to:

Now we can use the other property to combine them:

which simplifies to:
log(972)
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