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# Logarithmic equations: variable in the argument

## Video transcript

we're asked to solve the log of X plus log of 3 is equal to 2 log of 4 minus log of 2 so let me just rewrite it so we have the log of X plus the log of 3 is equal to 2 times the log of 4 minus the log of 2 or the logarithm of 2 and just as a reminder whenever you see a logarithm written without a base the implicit base is 10 so we could write 10 here 10 here 10 here and 10 here but for the rest of this example I'll just skip writing the 10 just because it'll save a little bit time but remember this little ready means log base 10 so this expression right over here is the power I have to raise 10 to to get X the power act to raise 10 to to get 3 now with that out of the way let's see what logarithm properties we can use so we know if we and these are all the same base we know that if we have log base a of B plus log base a of C then this is the same thing as log base a of B C and we also know so let me write all the logarithm properties that we know over here we also know that if we have a logarithm a log let me write it this way actually if I have B times the log base a of C this is equal to log base a of C to the B power and we also know and this is derived really straight from both of these is then if I have log base a of B minus log base a of C then this is equal to the log base a of B over C and this is really straight derived from these two right over here now with that other way let's see what we can apply so right over here we have all the logs of the same base and we have logarithm of X plus logarithm of 3 so by this property right over here the sum of logarithms with the same base this is going to be equal to log base 3 sorry log base 10 so I'll just want you to pull out and write it here log base 10 of 3 times X of the vx then based on this property right over here this thing can be rewritten let me say so this is going to be equal to this thing can be written as log base 10 log base 10 of 4 to the second power of 4 to the second power which is really just 16 so this is just going to be 16 and then we still have minus logarithm base 10 of 2 and now using this last property we know we have one logarithm minus another logarithm this is going to be equal to log base 10 of 16 over 2 right 16 divided by 2 which is the same thing as 8 so the right-hand side simplifies to log base 10 of 8 the left hand side is log base 10 of 3 X so if 10 10 to some power is going to be equal to 3 X + 10 to the same power is going to be equal to 8 so 3 X must be equal to 8/3 X must be equal to 8/3 X is equal to 8 and then we can divide both sides by 3 divide both sides by 3 you get X is equal to 8 over 3 one way this little step here I said look 10 to the this is an exponent if I raise 10 to this exponent I get 3 X 10 to this exponent I get 8 so 8 + 3 X must be the same thing one other way you could have thought about this is let's take 10 to this power on both sides so you could say 10 10 to this power and then 10 to this power over here if I raise 10 to the power that I need to raise 10 to to get the 3x well I'm just going to get 3x if I raise 10 to the power that I need to raise 10 to to get 8 I'm just going to get 8 so once again you got the 3x is equal to 8 and then you can simplify you get X is equal to 8/3