good day students in this clip we are going to be going over the top 5 common logarithm mistakes ok, so let's go ahead and take a look at the instructions, the instructions are as followed, we are to express the following as a single logarithmic expression where possible. alright so what i'm going to do here is i'm going to present the problem i'm going to try and simplify it using a common mistake and we'll try to see if you can figure out what the mistake is and then i'll go over what the mistake is ok? alright let's take a look at the first one, problem number one, what if you have the following expression : log base 3 of 5x multiplied by a log base three of 2x. so i can say to myself, since this is a product of logs two logs with the same base i can simply write this as log base 3 of 5x plus 2x alright? then that simplifies into log base 3 of 7x ok alright obviously this is the wrong answer. so do you see what the mistake is here? this is a misapplication of the property of logarithms. the product property of logarithms is as follows, let's go over that problem here. if your argument of your logarithmic function is a product of two factors xy then this can be rated as the sum of the factors. the log of the sum of the factors log base b of x plus log base b of y now if you look at this situation there are two separate logarithms the arguments are not being multiplied by each other this is a product of logarithms hence you can not apply the property here. ok? had it been something we had like log base 3 of 5x times 2x then you can apply this. it becomes log base 3 of 5x plus log base 3 of 2x ok? so the argument must be the product that's when you can apply a product property of logarithms. and this property is not used to condense it's basically used to expand as you can see this is a single logarithmic expression being written as two logarithmic expressions here. ok so just remember that the product property the arguments must be factors not the product of the two logarithmic expressions. so the answer here is um the correct answer what is the correct answer cannot be simplified, cannot be condensed alright? ok let's take a look at another mistake that students make on number two. what if you have um log base 3 of 15x divided by log base 3 of 10x. ok so some might think oh this is a you have a quotient of two logarithmic expressions here oh i can write this as a different i can write it as log base 3 of 15x minus 10x ok that simplifies into log base 3 of 15 minus 10, 5x of course this is wrong. now you see what the mistake is. lets go over the property the quotient property of logarithms. if you have, if the argument of the logarithmic expression is the quotient of two numbers alright,dividend and the divisor this can be written as um the difference of the log base the um the base the dividend minus the log of the divisor.so this can be written as the log of the numerator or the dividend minus the log of the denominator or the divisor. ok in order to apply this property the argument the logarithmic argument must be a quotient, is that the case here?absolutely not. in the numerator this log is not a quotient in the denominator this log is not a quotient, this is a quotient of two logarithmic expressions. hence this property cannot be applied here, ok? had it been you had something like this log base b of lets say you have 15x over 10x then you can write this as log base b of 15x minus log base b of 10x ok? now what you note here is that this property expands it doesn't consolidate it doesn't condense expressions alright? so hopefully you can see how this is a misapplication of this property there. so what is the correct answer to this problem? can it be condensed? the answer is no ok? the correct answer is the expression cannot be condensed due to a single logarithmic expression alright, this is another common mistake students make a misapplication of a quotient property. now let's take a look at property number 3 and common mistake number 3 that students make is, what if they have log base 2 of 3x multiplied by log base 2 of 3x, so is this two logarithmic expressions, these two logarithmic factors identical i can express this as log base 2 of 3x this entire logarithmic expression raised to the second power, and i can say oh wait i can use the power property here and express this as 2 log base 2 of 3x of course this is wrong. can you see what the mistake is? yeah this is a misapplication of the power property of logarithms. let's go ahead and take a look at what that uh property is real quick. so let's say you have um log base b of x to a power this is equal to you bring down that power , power times log base b of x ok? so if the logarithmic argument is an exponential expression, has a power, right? if the logarithmic argument has a power you can bring down that power as the, to become the coefficient
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