In Exercises 125–128, determine whether each statement is true...

Question

In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

$\frac{l o g_{7} 49}{l o g_{7} 7} = l o g_{7} 49 - l o g_{7} 7$

Accepted Answer

Hey everyone welcome back in this problem. We are asked to verify if the given equation is true. Okay and then if the statement is false, were asked to change the equation or amend the equation in order for it to be true. Okay so the expression or the equation we're given is log base 13 of 169 divided by log base 13 of 13 is equal to log base 13 of 169 minus log base of 13. Alright, so let's start with the left hand side. Okay let's determine if these are equal on the left hand side. We have log base 13 of divided by log base 13 of 13. Well, log base 13 of 13. That's just gonna be one. Okay let's deal with the top. Well we know we can write 100 and 69 13 squared. Ok. Writing that will allow us to use the exponent rules that we know. Okay, so log base 13 of 13 squared divided by log base 13 of 13. Alright, now we're calling that exponent rule log base B of X to the exponent N. Can be written as log N. Log. Base B of X. For the exponent comes out and can multiply the log Applying this to the numerator. We get to log B13 of divided by log base 13 of 13. Okay well log base 13 of 13 divided by log base 13 of 13 that's going to cancel and we're gonna be left with two. Now on the right hand side we have log base 13 of 100 and 69 minus log base 13 of 13. Doing the same thing again. We have log base 13 of 13 squared - Log Base 13 of 13. Using the exponent rule we get to log base of 13 -6 Base 13 of 13. And recall that log base B of B is just equal to one. Ok. So if the base and whatever's in the log are the same, that's equal to one. So this is going to give us two minus one which is equal to one. Ok, well the left hand side was to the right hand side was one. So those are not equal. Okay so scrolling down here we have the left hand side does not equal the right hand side. Okay, so this is a false statement. Okay well how can we correct this? Let's start with the left hand side. Let's start with what we had on the left hand side and see if we can try to find an expression that would be equal. So we have log base 13. of 169 Divided by Log Base 13 of 13. Let's recall. We have a change of base formula that says if we have log with base C. Of a divided by log base E. Of B. We can simplify these into one expression and it's going to be log base B. Of A. Okay so whatever is inside of the log in, the denominator becomes our base. Now whatever's inside the log and the numerator stays inside the log and this base of C. Is gone. We don't have to deal with C anymore. We apply this to our expression. Well we're gonna be left with log. The base is going to come from The denominator. Okay so our base is going to become And we're gonna have 169 inside. So This is a the original statement is false but we can rewrite it as log base 13 of 169, divided by log base 13 of 13 is equal to log base of 169. Okay. Thanks everyone for watching. See you in the next video.

In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
$\frac{l o g_{7} 49}{l o g_{7} 7} = l o g_{7} 49 - l o g_{7} 7$

Key Concepts

Properties of Logarithms

Quotient Rule for Logarithms

Evaluating Logarithms with Known Bases and Arguments

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