welcome to your tutor online video lessons my name is Joel and today I will teach you how to simplify logarithmic Expressions to think about logarithmic Expressions we just need to think of them as a way to undo variables that are in the exponent spot kind of like how we undo uh an a variable raised to the second power by applying the square root logarithms are especially useful for defining growth and Decay problems which will be the focus of another video to get started we want to make sure that we understand the identities and also the properties of logarithms so I'll Define those here properties Define um something on one side of the equation that is the same thing as the other here we want to make sure that we know log base B of B to some number is equal to that number uh when the bases are the same they will cancel out and take the log with it and we're left with that exponent okay so if we had log base 3 of 3^ squared the log and the 3es will cancel out and leave us with a two the same thing happens when we have a base with a log with the same base and as an exponent the logs and the bases cancel out and we're left with an X okay couple identities to make sure we know about identities are how we identi identities is how we get to zero and also one uh all we need to know is that the log with a base B of itself is equal to one and the log of anything of one is equal to zero log base anything of one is equal to zero okay now we're ready to get into a couple of examples when you simplify a logarithm the trick is to rewrite the number above the base using one of the identities or properties so that you can cancel out the logarithm we'll go ahead and look at two easy ones then we'll use um two complicated ones log base 3 of n we're going to want to rewrite the big number as a base three with an exponent so log base 3 how do we rate nine with a base of three well nine is the same thing as 3^ squ now we have the same bases and they cancel out and so log base 3 of 9 is the same thing as two okay one more easy example log base 4 of 64 again we're going want to rewrite the the big number as something else with the same base as we see below so log base 4 another four to the thir power the logs and the four cancel out and we're left with three see how that works all we're doing is taking the big number and rewriting it as um this base to some exponent if you can figure out that exponent with you have your simplification all right let's look at two little more involved problems excuse me okay sorry about that uh let's look at our a little bit more involved problem we have log base 5 of 1 over 25 again we're going to take that whole thing and we're going to rewrite it as five to some power 1 over 25 I know I'm looking for a negative exponent so go ahead and rate it my five a negative exponent well to get to 20 I know it's a two -2 cancel these out and we get -2 all right one more really involved problem and then I think we've covered almost all the simplification uh types for logarithms all right log base 3 of 9 times the 4th root of 1 27 okay lots of different parts going on with this one but we're looking for them all to be base three go ahead and rewrite my log and base and I can do each part individually the nine is the same thing as 3 squ and I'll go ahead and break this down step by step 1 over 27 is the same thing as 3 to the3 go ahead and get another simplification step here keep everything as base threes it's kind kind like we're unsimplified in order to get the same base now uh fourth root is a fractional root and I know we have -3 over 4 and now we add our exponents to plus-3 4 log base 3 of 3 to the 54s is our exponent the logs and the bases cancel out and we're left with our answer 54 fourths well thanks for joining me for this lesson I hope it was helpful for you help me help you by leaving comments and asking questions you can email me Joel atyt online.com and you can go over and like the Facebook page and um ask your questions and comments and video suggestions there and if these videos are helpful to you please share them with your classmates I'll see you next time happy studying