in this lesson we're going to focus on evaluating logs so what is log base 2 of 16 what is that equal to 2 raised to the what power is 16. how many twos do you have to multiply to get to 16 2 times 2 times 2 times 2 4 times 16 so therefore 2 to the fourth is 16. so log base 2 of 16 is equal to four so now that you know that what is log base 3 of 27 how many threes do you need to multiply to get to 27 you need to multiply 3 3 times it gets 27 3 times 3 times 3 is 27 so log base 3 of 27 is 3. now what about log base 5 of 25 how many fives do you have to multiply to get to 25 you need to multiply two fives five squared is 25. now what about log base four of one log of one is always equal to zero what about log base seven of seven to the first power is seven so if these numbers are the same they will cancel it's simply one now what is log of a thousand what should you do if you're not given a base if there is no base it's always assumed to be a ten so how many tens do you have to multiply together to get to a thousand you need to multiply ten three times this will give you three zeros and that's equal to a thousand so ten to the third is one thousand therefore log base ten of a thousand is three now what about log base 10 of 100 10 times 10 is 100 so you need to multiply 210 to get to 100 so it's two now what about log of zero .0001 what is that equal to so the base is ten it turns out that ten to the negative four is point zero zero zero one so this is equal to negative four so here are some things to know log of one is always zero log of ten when the base is ten this is equal to one notice a pattern log of a hundred is equal to two and log of a thousand is equal to three notice that when you have two zeros it's two log thousand one thousand has three zeros this stream so based on that what is the log of one million notice that there are six zeros this is going to equal six log of ten thousand has four zeros so this is equal to four now what about log of point one this is negative one log of point zero one is going to be negative two log of point zero zero one that's going to equal negative three and log of point zero zero zero one is equal to negative four assuming of course the base is ten which it is if no base is written so keep that in mind now let's work on some other examples let's say if you were to see something that looks like this on the homework what do you think the answer is what is 7 log base 7 of 38 equal to whenever you see this simply cancel the 7 and it's going to equal to whatever you see here which is 38 so knowing that try these two what is 5 log base 5 of 14 and also 8 log base 8 of y so this is going to equal 14 and this one is simply going to equal y now let's try some more examples what is log base 3 of 9 equal to well we know we have to multiply two threes to get to nine three squared is nine so the answer is two now what is log base nine i mean base three of one over nine how will that change the answer it turns out that this is going to be equal to negative two three squared is positive nine three to the negative two you need to flip the fraction is gonna be one over nine the negative exponent will cause the nine to move from the top to the bottom now what if you reverse the numbers if log base 3 of 9 is 2 what's a log base 9 of 3 so here's a hint there's going to be a 2 involved it turns out that it's 1 over 2 you need to flip the fraction if you reverse three and nine and finally what is log base nine of one over three this is going to be negative one over two so anytime you have a fraction notice that it's going to be negative anytime this number is larger notice that you'll have a fraction now if this number is larger than this one not including the fraction if you just compare 93 then the answer typically will be a number that's greater than one it's not going to be a fraction let's work on some more examples these log base 2 of 32 log base 2 1 over 32 log base 32 of 2 and log base 32 1 over 2. now what is log base 2 of 32 how many twos do you have to multiply to get to 32 it takes five twos to get to 32 so this is five now if two to the fifth is 32 that means two raised to the negative five is one over thirty two so this is going to be negative five now 32 raised to the what power is two it turns out that the fifth root of 32 is two so this is going to be one over five now if 32 raised to the one over five is two thirty two to the negative one over five is one over two and so that's this answer so there's going to be a 5 somewhere it's either positive 5 or negative 5 positive 1 over 5 or negative 1 over 5. and if you convert it to its exponential form it can help you to determine which answer is correct so let me give you some practice problems but mix in the order go ahead and find the answers to these questions two to the what power is eight two to the third power is eight two times two times two three times eight now we know that three squared is nine but three raised to the negative two is one over nine so that's negative two the square root of 25 is five so 25 raised to the one half is five the square root of 64 is eight so 64 raised to the one half is eight now we know two to the fourth is 16. so the fourth root of 16 is two so 16 raised to the one fourth is two therefore sixteen raised to the negative one fourth is one over two now three to the fourth is eighty-one so the fourth root of eighty-one is three so if 81 raised to the 1 4 is 3 81 to the negative 1 4 is 1 over 3 and so that's this you