In Exercises 21–42, evaluate each expression without using a calc...

Question

In Exercises 21–42, evaluate each expression without using a calculator. log5 (1/5)

Accepted Answer

Hey everyone welcome back in this problem. We are asked to evaluate the following expression and the expression we're given is log base nine of 1/9. Okay so let's go ahead and write what we're given log base nine of 1/9. Okay, we're going to set that equal to why and why is going to be what we're looking for? Okay, that's gonna be the value of that expression evaluated. Alright, so let's recall our log rules. Well we have a log rule that says if we have log with base B. Of some quantity X. And that's equal to y. Okay just like in our case equal to Y. But we can rewrite that as B to the exponent Y is equal to X. So the base of the log becomes the base of the exponent. Alright, so applying that to our expression, we have base nine so that's gonna be nine is going to become are the base of our exponents. We're gonna have the exponent wide and on the right hand side we're gonna have 1/9. Okay, so nine to the Y. Is equal to 1/9. All right now when we have an equation like this, what we want to do is get the base on both sides to be the same. Okay, if the base is the same on both sides we can compare the exponents. So on the left hand side we have nine to the Y. Let's try to get a base of nine on the right hand side. Well let's use our exponent rules recall if we have one over some em to the exponent N. We can write this as m to the minus end. Now if we apply that to the right hand side, what we're going to have is we're just gonna have nine to the -1. Alright, so now we have an equation, we have the same base on both sides, Okay? And they're equal. That means that the exponents must be equal. Okay? So this tells us that why is equal to minus one? Okay, and why was the value? We were looking for the evaluation of that expression? Okay, so our answer is going to be d minus one. I hope this video helped. Thanks for watching everyone.

In Exercises 21–42, evaluate each expression without using a calculator. log5 (1/5)

Key Concepts

Logarithms

Change of Base Formula

Properties of Logarithms

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