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

Question

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

$\log_{5} 4^{6} = 4 \log_{5} 6$

Accepted Answer

Welcome back, everyone determine the validity of the given statement log base eight of 12 to the power of seven equals 12 multiplied by log base eight of seven. We're going to answer choices A says true and B says false. So for this specific problem, because we have a power, we're going to use the power rule. Let's suppose that we have a log of base A of number B raise the power of M, then we can rewrite this as M multiplied by log of the same base A of the number B, we're essentially bringing that exponent down. So if we look at our expression on the left hand side, since we have log of base eight of 12 to the power of seventh, well, we can essentially bring that seven down and it essentially becomes the leading coefficient. So we get seven multiplied by log of base eight of 12. Now this is our simplified expression. And the statement says that it must be equal to 12 log 87, right? So now we can see that we have exactly the same basis. So we don't need to convert our basis. We essentially are looking at log of eight log base eight of 12 being equal to log base eight of seven, right. And we have completely different leading coefficients. Those two expressions are not equal to each other. There is no property that allows us to convert our logs such that we swap the leading coefficient with the value of the log, right. So essentially the given statement is going to be false. And the correct answer to this problem is option B. Thank you for watching.

Determine whether the following statements are true and give an explanation or counterexample.
$\log_{5} 4^{6} = 4 \log_{5} 6$

Key Concepts

Logarithmic Properties

Change of Base Formula

Exponential Equations

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