Problem

What does the change-of-base formula do? Why is i…

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Problem 1 Easy Difficulty

How does the power rule for logarithms help when solving logarithms with the form $\log _{\mathrm{b}}(\sqrt[n]{x}) ?$

Answer

$\log _{b}(7)+\log _{b}(x)+\log _{b}(2) \log _{b}(y)$

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Algebra

College Algebra

Chapter 6

Exponential and Logarithmic Functions

Section 5

Logarithmic Properties

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Video Transcript

Hello. Welcome to the video. And should anyone be taking a look at logger rhythms? And specifically, how can we use the properties of longer rhythms to rewrite them and simplify them? This case, we are given this longer than here. Log base B of the end, Fruit X, and we're told to use the power rule to simplify this. We're gonna solve this in two steps here. Ah, step one. We're gonna remind ourselves what exactly the power rule of law algorithms is. So if we're given a longer of them that looks like this, a log base B B can be any value here. Doesn't really matter of any value a raised to the power off. Let's just say see, in this case, we can rewrite this as the exponents see time's log base Be even. Mind that this the same base value off whatever this value is here. So a and keep in mind, if we see a longer than like this, we can write it like this and vice versa. And so that's one thing. And then Step two, we can look at what's inside the parentheses and write them and rewrite it as an expression that would look a little more familiar in terms of the power so we can rewrite the end fruit of X as X to the power of one over end. These are interchangeable. These both equal the exact same thing there. They're telling us the exact same thing. So if we I love this into the log a rhythm, uh, the log base B of the n through it of X equals in other terms, the log base B of X to the one over end. And this It's something we can use the power ruling. So going back to what we're given Log base B of X to the one and power you can take this exponents put it out front and multiply by log base. In this case, be of what's on bottom here, X So one time. One over end times the law based be of X or simplifying that the log base, B X, over end. And this is our final answer

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