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Numerade Educator



Problem 37 Hard Difficulty

Find the exact value of each expression.

(a) $ \log_{10} 40 + \log_{10} 2.5 $
(b) $ \log_8 60 - \log_8 3 - \log_8 5 $


a) 2
b) $\frac{2}{3}$

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Video Transcript

okay to simplify and evaluate these expressions were going to use some properties of logarithms. So we have the product property here, and that tells us that we can take a log plus a log and change it to the log of the product so we can change the law. Based 10 of 40 plus the log based 10 of 2.5 into the log based 10 of 40 times 2.5 and 40 times 2.5 is 100. So we have the log based 10 of 100 and that means 10 to 1 10 toe. What power is 100 that would be too now for Part B will involve the quotient property. So the log of something minus the log of something else is the log of the quotient, so we can use that here because we are subtracting logs. Another thing we can do is we can factor out the negative first from the second and third terms so long. Based eight of 60 minus the quantity log based eight of three plus law based eight of five a factor, the minus from before both of those logs, and we can start by using the product property from Part A on the second part of that problem. So we have Law based eight of 60 minus log based eight of 15. Now, according to the quotient property, we can change it into law based eight of 60 divided by 15. So we've logged based eight of 75th which is the same Islam based eight of four. Let's see if we can simplify this anymore because there's definitely a relationship between eight and four. Force the power of to and so is eight. So let's think of this as log based. Eight of two squared. Now we have our power property of logarithms that says you can bring this power out to the front. So this is equivalent to two times Log based eight of two. Do we know what long base eight of two is? A tow? What power is too Well, eight to the 1/3 power would be too, so we have two times 1/3. So the answer is 2/3