# Rewrite as a single logarithm and simplify fraction

So we can simplify. Rule 6: Log of Exponent Rule The logarithm of an exponential number where its base is the same as the base of the log equals the exponent.

I can put together that variable x and constant 2 inside a single parenthesis using division operation. Log base 5 of 25 to the x over y using this property means that it's the same thing as log base 5 of 25 to the x power minus log base 5 of y.

So this part right over here can be rewritten as x times the logarithm base 5 of We have to rewrite 3 in logarithmic form such that it has a base of 4. And this is useful because log base 5 of 25 is actually fairly easy to think about.

## Log calculator

The square root can definitely simplify the perfect square 81 and the y12 because it has an even power. I'll do this in blue. Other textbooks refer to this as simplifying logarithms. Now, this looks like we can do a little bit of simplifying. So this part right over here can be rewritten as x times the logarithm base 5 of And I do agree that this does require some simplification over here, that having this right over here inside of the logarithm is not a pleasant thing to look at. Unnecessary errors can be prevented by being careful and methodical in every step. Just a reminder, you add the exponents during multiplication and subtract during division.

And then, of course, we have minus log base 5 of y. But, they all mean the same.

### Single logarithm examples

Rule 6: Log of Exponent Rule The logarithm of an exponential number where its base is the same as the base of the log equals the exponent. Just a reminder, you add the exponents during multiplication and subtract during division. And we're done. Rule 7: Exponent of Log Rule Raising the logarithm of a number by its base equals the number. The square root can definitely simplify the perfect square 81 and the y12 because it has an even power. And here we have 25 to the x over y. Example 4: Combine or condense the following log expressions into a single logarithm: I can apply the reverse of Power rule to place the exponents on variable x for the two expressions and leave the third one for now because it is already fine. Remember that Power Rule brings down the exponent, so the opposite direction is to put it up. This part right here is asking us, what power do I have to raise 5 to to get to 25? Other textbooks refer to this as simplifying logarithms. So then we are left with, this is equal to-- and I'll write it in front of the x now-- 2 times x minus log base 5 of y. Rule 2: Quotient Rule The logarithm of the quotient of numbers is the difference of the logarithm of individual numbers. I'll do this in blue.

Rule 2: Quotient Rule The logarithm of the quotient of numbers is the difference of the logarithm of individual numbers. This is how it looks when you solve it.

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