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Properties of Logarithms - Concept

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Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

In order to solve equations with logarithms and exponential functions, we must understand two properties of logarithms. These properties of logarithms are the methods we use to get to and solve for variables that are in exponents or inside logarithms. The two properties of logarithms say that we can use exponentiation and logarithms to cancel each other.

Hi, the whole idea behind the logarithmic functions is that they're inverse of exponential functions. So if f of x equals b to the x then f inverse of x is the log base b of x. And that gives rise to 2 identities. First f inverse of f of x which is log base b of b to the x is x. And second f of f inverse of x which is b to the log base b of x also equals x. Now this second identity is only true for positive values of x because only positive values of x are in the domain of the log base b. So this second one is only true for x greater than 0 but this identity is true for all x.
Let's use these identities in some exercises, says evaluate each of the following log base 2 of 32 well first you've got recognize that 32 is a integer power of 2, log base 2 of 2 to the fifth and once you see that you can use the first identity log base b of b to the x equals x. So log base 2 of 2 to the fifth is 5. Here's another one log base 5 of 0.004 and here I've kind of disguised it a little bit the 0.004 is going to be a power of 5, it's the same as 4 over 100 which is the same as 1 over 25 and 1 over 25 is 5 to the negative 2. And again by the first identity log base 5 of 5 to the negative 2 is just 2.
This one is a little tougher, certainly 2 in not an integer power of a somehow we'll make the identities work. We've log base 8 of the cube root of 8 right 2 is the cube root of 8 and the cube root of 8 is 2 to the one third. Log base a of 8 to the one third is one third. Remember the common logarithm, whenever you see a log without the base written log this is the base 10 log and so this is the same as the log base 10 of 10 to the sixth and so that's just 6. Same thing here, this is the common log, log base 10, so this is log base 10 what power of 10 is this? It's the square root 10 cubed so that's 10 to the three halves, so again log base 10, 10 to the three halves is three halves.
One last example, I didn't use this second identity until now but now I need the second one, trouble is these are not the same bases so I'm going to have to do something here. I can write 9 as 3 squared, 3 squared to the log base 3 of x and I can use the power to a power property 3 to the 2 log base 3 of x and then I can use the log of a power property this coefficient can come inside as an exponent I get 3 to the log base 3 of x squared. So by the second identity 3 to the log base 3 of x squared is just x squared.

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