hey everybody this is Paul today I'm going to be showing you how to expand a logarithm so this is lesson 10 in my tutorial on logarithms and up till now I've been showing you properties of logarithms and now I'm going to show you how to use those properties to expand an expression so if we have log base 3 of x^2 y 3r / Z and we want to use properties of logarithms to stretch out the statement to make it basically a longer form we can use the property I've been talking about in my previous videos so one of the properties I explained to you is that when we have a logarithm of a fraction then that's equal to the logarithm of the numerator minus the logarithm of the denominator so let's go ahead and write that out here so log base 3 of all this stuff is equal to log base 3 of the numerator so our numerator is what's on top x^2 Y 3r and minus log base 3 of our denom denominator and the denominator is what's on bottom which is our Z so we've expanded this a little bit but we can use some of the other properties to expand this a little more so another property that I told you about was when you have a logarithm of a product two things multiplied by one another then that can be split up into the logarithm of the first plus the logorithm of the second so let's go ahead and do that so we're going to break this apart into log base 3 of the first which our first is X2 plus log base 3 of the 2 in this case it's y 3 and this last term is simplified so we're just going to go ahead and rewrite it and so now we have another property we can use um one of the things I told you earlier was that you can take an exponent and bring it out front so if you have some number raised to some power you can bring that exponent out front of the logarithm so let's go ahead and do that for both of these terms so this is equivalent to sorry this is equivalent to bringing the two out front and then just writing the rest so this is equivalent to 2 log base 3 of X we just move the two out front and everything else stays the same and we're going to go ahead and do that to this guy so bringing the three out front this becomes 3 log base 3 of Y and then this guy doesn't change log base 3 of Z so by using properties of logarithms that I've explained in my previous videos we found that log base 3 of x^2 y 3r / Z is equivalent to 2 * the log base 3 of x + 3 * the log base 3 of Yus log base 3 of Z so anyway thanks for watching um hopefully you guys have learned some stuff from this and uh you guys have a great day and stay tuned for another example I'll do a couple more examples of this stuff so stay tuned for that and uh don't forget to subscribe