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 * 2 * 2 * 2 4 * 16 so therefore 2 4 is 16 so log base 2 of 16 is = to 4 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 * it gets 27 3 * 3 * 3 is 27 so log base 3 of 27 is three 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 5^ s is 25 now what about log base 4 of 1 log of one is always equal to zero what about log base 7 of 7 7 to the first power is s so if these numbers are the same they will cancel it's simply one now what is log of a th000 what should you do if you're not given a base if there is no base it's always assume to be a 10 so how many T do you have to multiply together to get to a th you need to multiply 10 three times this will give you three zeros and that's equal to a th000 so 10 3 is 1,00 therefore log base 10 of a th000 is three now what about log base 10 of 100 10 * 10 is 100 so you need to multiply two 10 to get to 100 so it's two now what about log of 0.1 what is that equal to so the base is 10 it turns out that 10 to the4 is 0.1 so this is equal to4 so here are some things to know log of one is always zero log of 10 when the base is 10 this is equal to 1 notice the pattern log of 100 is equal to 2 and log of a th000 is equal to three notice that when you have two zeros it's two log th 1,00 has three zeros is three So based on that what is log of 1 million notice that there are six zeros this is going to equal six log of 10,000 has four zeros so this is equal to four now what about log of 0.1 this is negative 1 log of 01 is going to be -2 log of 001 that's going to equal -3 and log of .001 is equal to4 assuming of course the base is 10 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 seven and it's going to equal to whatever you see here which is 38 so knowing that try these two what is five 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 2 well we know we have to multiply two threes to get to 9 3^2 is 9 so the answer is two now what is log base 9 I mean base three of 1 over 9 how will that change the answer it turns out that this is going to be equal to -2 3^2 is POS 9 3 to the -2 you need to flip the fraction it's going to be 1 over9 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 log base 9 of 3 so here's a hint there going to be a two involved it turns out that it's 1/ two you need to flip the fraction if you reverse 3 and n and finally what is log base 9 of 1 over3 this is going to be negative 1/2 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 9 and three 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 try these log base 2 of 32 log base 2 1 over 32 log base 32 of 2 and log base 32 1/ 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 2 to the 5th is 32 that means 2 ra to 5 is 1 over 32 so this is going to be -5 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 1 over 5 now if 32 rais to the 1 over 5 is 2 32 to the 1 over 5 is 1 over two and so that's this answer so there's going to be a five somewhere it's either positive five or neg five positive 1 over five or negative 1 over5 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 8 2 the 3 power is 8 2 * 2 * 2 3 * is 8 now we know that 3^ 2 is 9 but 3 ra to the -2 is 1 over 9 so that's -2 the square OT of 25 is 5 so 25 raised to the 12 is 5 the square root of 64 is 8 so 64 raised to the 12 is 8 now we know 2 to 4 is 16 so the fourth root of 16 is 2 so 16 raised to 1/4 is 2 therefore 16 rais to the 1/4 is 1/ 2 now 3 to 4 is 81 so the four root of 81 is 3 so if 81 raised to the 1/4 is 3 81 to the 1/4 is 1 over 3 and so that's this for