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For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.$$\ln \left(\frac{1}{4^{\mathrm{k}}}\right)$$

$-k \ln (4)$

Algebra

Chapter 6

Exponential and Logarithmic Functions

Section 5

Logarithmic Properties

Functions

Baylor University

Idaho State University

Lectures

01:32

In mathematics, the absolu…

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01:17

For the following exercise…

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02:57

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Hello. Welcome to the video. And today we're gonna be taking a look at lawns and Natural in this case, were given the natural log of one over four to the power of King. This problem tells us to expand this as much as we can. Now, how we do this, we'll need to know our log properties. And one thing that's important to know, especially in this case, is that log properties can also be applied to national solve. This problem will need two of these properties the motion rule and power. Now the ocean rule reads, if you have a law that looks like this so log base, any value, be off any value a over you see, it could be rewritten as the log of B. And keep in mind, the base value here does stay the same off whatever's on top, minus the lock base B of whatever's aren't Bob. And keep in mind for all of these rules, you can, uh, you can use these interchangeably where you can change this into this. You can change this into this. That's the portion rule, and the power rule reads that if you have a log base B off in this new let's say we have four to the power off value. Kay, we can rewrite this as the exponents Kay times the log base be off. Whatever the base of the ex opponent here is so four. And once again you can interchange these and rewrite them is teach as each other. So going to the problem were given natural log of one over four to the power of K. We can use the quotient rule to rewrite. This is the national log one minus the natural laws of force The power of K and we can keep going a cz we know the natural log of one is zero and we can use the power rule for this one to rewrite it as k times the natural log of the base here for And we can just do simple subtraction here to get our final answer off Negative K times The natural law four

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