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)
(