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Combine the given expression into a single logarithm.$$\frac{1}{4} \ln x+\frac{2}{3} \ln y+3 \ln z-4 \ln w-\frac{2}{5} \ln v$$

$$\ln \frac{\sqrt[4]{x} \sqrt[3]{y^{2}} z^{3}}{w^{4} \sqrt[5]{v^{2}}}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

Missouri State University

Harvey Mudd College

Lectures

02:37

Combine the given expressi…

02:15

01:03

Write the given expression…

01:02

Condense the expression to…

01:54

condense the given logarit…

01:15

01:23

01:22

00:48

Write the logarithms as $a…

02:33

00:33

Write each expression as a…

00:35

Write the following expres…

01:37

Write the expression as th…

01:26

Simplify the following exp…

Okay, so we're asked to simplify this problem, and we're just gonna use a couple law of logs and some exponents tricks? Why, Mrs three Natural Z I did not give myself much room, but hopefully you can see all my my answers on. I have a preference in doing this. You don't have to do the problem my way. Okay, But I like my way. Because then I only have to remember one to law of logs. The first one is the all the numbers that are in front can be rewritten as exponents. And I'm gonna stick these negatives attached to those values. So, I mean, I make negative four become the exponents, and then I'll change that to a plus sign. And I gotta do that only as well. So I'm looking at natural log of X to the 1/4 plus natural log of why to the two thirds plus natural log of Z Cube and then plus E. I said earlier to the negative fourth power and then plus natural log of the two faced power. Eso I think at this point I could do two steps at once. Is any time you see a plus sign, we can combine them using multiplication. And the other rule that I'm gonna have is that you understand that to the 1/4 power is the fourth root. Um, every every student capitalist should know that. So the plus sign, I mean, combine them as a single log on day. This is the cube root of why squared Cuba goes into the index at times. Uh, Z Cube has already, as an ex moment times w to the negative fourth times. The e did not. I did not do that. Um, so I mean, leave that as a negative exponents, but it is the fifth root of that eso Why do I do it that way? Is because then anytime you see a negative exposure, you could say Oh, yeah, negative exposure goes into the denominator. That way you don't have to remember all the other rules. Well, there's only one other rule, and that's when you subtract you divide, which is the same thing as I what I just did. But this year, correct answer circled in green

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