In Exercises 1–40, use properties of logarithms to expand each lo...

Question

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.

log4 (64/y)

Accepted Answer

Hello. Today, we're going to be using the properties of logarithms to simplify the given logarithms. So the given logarithms is the log base of three of 81 divided by K. Now before we start using the logarithms, let's just go ahead and quickly rewrite 81 81 can be rewritten as three to the power of four. So we can go ahead and rewrite this logarithms as log base three of three to the power of four divided by K. Alright, if we look at the inside argument of our longer rhythm, it's using division. So let's go ahead and use the division property for logarithms to rewrite this. And the division property states that if you have the longer rhythm of A over B, this could be rewritten as the log base of A minus the log base of B. So we're gonna go ahead and use the division property for logarithms to go ahead and rewrite this logarithms. So A. Is going to equal to three to the power of four, and B is going to equal to K. So using this property we can rewrite this as the log base of three of three to the power of four minus the log base of three of K. All right now we don't want to have this exponents inside the logarithms. So let's go ahead and use the exponential property for logarithms to rewrite the left hand side. And that states that if you have the logo rhythm of A to the power of B. You can bring forward the exponent in front of the logo rhythm as a product or in other words the log base of A. To the power of B can be rewritten as B times the logarithms of A. And using this property we can rewrite the left hand side as four times the logarithms of base three of three minus the logarithm of base three of K. Alright, and there is one more simplification that we can do recall that if you have the logarithms whose base matches its argument, you can go ahead and rewrite that value as one. Here we have the log base three of three, so the base matches the argument. So this logarithms can be rewritten as one and four times one is just going to give us four. So using the exponential property, we can rewrite this as four minus the logarithms of base three of K. And this is going to be the complete simplification of our given logarithms and that means that the answer to this problem is going to be be So I hope this video helps you in understanding how to use the properties of logarithms and I'll go ahead and see you all in the next video

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log4 (64/y)

Key Concepts

Properties of Logarithms

Change of Base Formula

Evaluating Logarithmic Expressions

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