in this video we're briefly going to go over some properties of logs that you need to be familiar with the first one is the power rule let's say we have log a raised to the n you can move the exponent in front so this is equal to n log a the second is the product rule log a times log b is equal to log a plus log b and then you have the quotient rule log a divided by log b is log a minus log b so we can use this in order to evaluate logs so for instance let's say if we want to simplify log base 5 of 5 raised to the 7. so what we can do is move the seven to the front so this is equal to seven log base five of five a log base five of five because they're the same it's equal to one and 7 times 1 is 7. let's try another example like that log base 2 of 8 raised to the fifth power so let's move the 5 to the front so this is equal to 5 times log base 2 of 8. now how many twos do we need to multiply to get to 8 we need to multiply 3 2s 2 to the third is 8 so log base two of a is three and five times three is fifteen now let's work on this example log base two of sixteen times eight so we can use the product rule to separate the 16 and the 8. if you recall log of a times b is equal to log a plus log of b so therefore log base 2 of 16 times 8 is going to be log base 2 of 16 plus log base 2 of 8. now log base 2 of 16 that's 4 because 2 to the fourth power is sixteen log base two of eight is three so the final answer is four plus three which is seven try this one log base three 27 times 81 so first let's separate it into two logs this is log base 3 of 27 plus log base 3 of 81 3 to the third power is 27 so log base 3 of 27 is 3 log base 3 of 81 is 4 because 3 to the fourth power is 81 and this adds to 7. now what about division what is log base 4 of 256 minus 64. so using the quotient rule log a divided by b is equal to log a minus log b so therefore this is going to be log base 4 of 256 minus log base 4 of 64. 4 to the fourth power is 256 and four to the third power is 64. and four minus three is one and so that's going to be the answer try this one log base two of 128 over 8. so this is equal to log base 2 of 128 minus log base 2 of 8. now 2 to the 7th power is 128 and 2 to the third is 8 so 7 minus 3 is 4 and that's going to be the answer here's the last example log base 2 of 128 times 64 divided by 8 times 16 raised to the fifth power so the first thing we should do is move the five and it's going to be distributed to everything now every number that's on top will have a positive sign the numbers on the bottom will have a negative sign so it's going to be log base 2 of 128 plus log base 2 of 64 since they're multiplied to each other minus log base 2 of 8 since it's divided by and also minus log base 2 of 16. log base two of one twenty eight we know it's seven two to the sixth power is sixty four two to the third is eight two to the fourth is sixteen negative 3 and negative 4 adds up to negative 7 which cancels with the positive 7. so the final answer is 5 times 6 which is 30. you