In Exercises 41–70, use properties of logarithms to condense each...

Question

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator.3 ln x + 5 ln y - 6 ln z

Accepted Answer

Hello everyone in this video. We're going to be looking at this practice problem where we want to condense the equation two times natural log W plus nine times natural log B minus four times Natural log you into a single algorithm with a coefficient of one. And we want to try to condense or evaluate without using a calculator. So in order to evaluate this without using a calculator and it probably wouldn't even be possible with a calculator because we have all these variables. We want to remember the logarithmic properties. So say we have two natural logs being added together, such as natural log A plus natural log B. We can condense them into a single algorithm because there is addition between them. I'm going to multiply the values together. So natural log A plus natural log B becomes natural log A. B. And recall that that one is. This property is known as the product rule and say I had natural log a minus natural log B. I can combine into a single algorithm instead of using multiplication. I'll use division where a natural log a minus natural log B becomes natural log A divided by B. And that one is known as the quotient rule. And a final rule that we're going to use or is helpful in condensing logarithms and trying to get a coefficient. One is the power rule and the power rule is for when you want to get rid of a coefficient. So so you have coefficient A in front of Natural log B. To get rid of the A. We can make an exponent of B. So eight times natural log B becomes natural log B to the power of A. So we're going to use these rules to try to condense our equation into a single algorithm. So if we take it by parts, you can see that all of these natural logs have coefficients in front of them. So the first thing I'm going to do is apply the power rule to all of the coefficients. So if we look at this first part, I have two natural log W plus nine. Natural log v minus four. Natural log you so I can move the two to be power of W. So now I have natural log W squared If I do the same to this, I have natural log v to the 9th power minus natural log U to the fourth power. So notice how I just applied the power rule to get rid of all the coefficients. So now I don't have any more coefficients, but I still have three logarithms. So I'm going to start by combining the first two and notice I have addition. So I'll be using the product rule. So this becomes natural log w squared times V to the ninth. So notice how I combine the two logarithms. Using the product rule by multiplying w two and V nine and I still have the second algorithm, but now I have subtraction. So I'll be using division or the quotient rule. So, natural log w squared times V. To the ninth minus. Natural log U. To the fourth becomes natural log V W squared times V. To the ninth, Divided by the 2nd value, which is U. to the 4th. And you can see that I now have only one logarithms with coefficient one, so this becomes my final answer, and this aligns with answer choice, answer choice deed. So hopefully this video helped and see you next time.

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator.3 ln x + 5 ln y - 6 ln z

Key Concepts

Properties of Logarithms

Condensing Logarithmic Expressions

Natural Logarithm (ln)

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