Determine whether the following statements are true and give a...

Question

Determine whether the following statements are true and give an explanation or counterexample.

$\frac{\log_{b} x}{\log_{b} y} = \log_{b} x - \log_{b} y$

Accepted Answer

Welcome back. Everyone. In this problem, we want to determine the validity of the given statement, log base Z of A divided by log base Z of B equals log base Z of A minus log base Z of B A says it's true. B says it's false. Now, if we're going to determine if the statement is valid, then we need to evaluate what we have on the left hand side of our statement and see if it gives us what we have on the right hand side. So let's start there. We're dividing the log of sorry log base Z of A, OK by log base Z of B. No, there's a rule of logarithms that looks a little bit like this, which would give us our statement on right. OK. But it's not exactly the same thing. OK. Recall that in logarithms, if we are finding the log of a quotient, OK. So let's say the log base X of A divided by B, then we could rewrite it as log base X of A minus log base X of B. But here we're not finding the log of a quotient. We're dividing two logs entirely on their own by each other. So the law or the rule here would not apply to this situation. In other words, log base Z of A divided by log base Z of B would not be equal the log base Z of A minus log base Z of B. Therefore, this statement is false. Thanks a lot for watching everyone. I hope this video helped.

Determine whether the following statements are true and give an explanation or counterexample.

$\frac{\log_{b} x}{\log_{b} y} = \log_{b} x - \log_{b} y$

Key Concepts

Logarithmic Properties

Change of Base Formula

Counterexamples in Mathematics

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