Condensing Log Expressions

Since these logs have the same base, the Product Rule says that the addition outside the two logs can be turned into multiplication inside one log:

Then the answer is:

Since these logs have the same base, the Quotient Rule says that the subtraction outside the two logs can be turned into division inside one log:

Then my answer is:

The Power Rule says that the multiplier out in front of the log can be turned into an exponent on the insides of the log:

Then my final answer is:

Using the Power Rule, I will get rid of the multipliers outside the logs by moving them inside the logs as powers:

Then I'll move the terms in order to put the added terms together, leaving the one "minus" term to the end of the string:

I'll apply the Product Rule to convert the addition between the first two log terms into multiplication inside one combined log term:

Then I'll account for the subtracted term by applying the Quotient Rule to move it inside, dividing the insides of the other term:

Then my final answer is:

How the heck am I supposed to combine the 5 with the log?

While the Log-Cancelling Rule is usually applied to simplify away from logs, I can use it to go the other way; in other words, I can apply the Log-Cancelling Rule to go backwards from the number to a log expression. This Rule gives me:

This Rule is almost never used this way, but it does what they want of me; namely, it converts a plain number into a log expression, which I can then combine (via the Product Rule) into one log. Just in case they want the numeric part evaluated, I'll turn the 7⁵ into 16807:

I don't know if they want the 7⁵ to be multiplied out, so I'l include both forms in the answer, just to be safe:

The one-half that is multiplied onto the first log can be taken inside (by the Power Rule) as an exponent:

The one-half power means the same thing as a square root, so I get:

Now I can combine the two logs by using the Quotient Rule:

I don't see anything more I can do with this, so my hand-in answer is: