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 - (1/3) ln y

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 c minus one half natural log D into a single logarithms with a coefficient of one. And we want to do that without using a calculator. So in this case because we're combining logarithms and we're trying to condense logarithms, it's important to remember the properties for condensing logarithms. So in this case we have two logarithms, there's a subtraction between them and they also have coefficients that are not one. And recall that we want a coefficient of one. So we need a way to get rid of these coefficients and combine them even though they're being subtracted. So in this case say we have logarithms A minus logo rhythm B. We can combine the two using the quotient rule, which states that when you have subtraction between two logarithms, you can combine them using division. So natural log A minus natural log B become natural log A divided by B. So notice how I came in with two natural logs and I came out with one and that's what we were trying to do in this problem. Another rule to remember is the power rule and the power rule applies when we have coefficients in front of the natural logs that we want to get rid of. So say we have a times natural log B. We can get rid of the A. By making it an exponent of B. So a times natural log B becomes natural log B to the power of A and notice how the coefficient here is now one. So we're going to use both of these rules to condense our equation. And if we do that, I'm going to start by using the power rule because both of these natural logs have coefficients where my first Natural log has coefficient to in my second natural log has coefficient 1/2. So if I apply the power rule, I have two natural log C minus one half. Natural log D. If I apply the power rule, I'm going to move the two to be an exponent of C. So I'll have natural log C squared minus, I'll move the one half to be a power of D. So I'll have natural log D to the one half. And recall that D to the one half or anything to the one half is the same as taking the square root. So I can rewrite this as natural log c squared minus natural log square root of D and notice now I only have coefficients of one, but I still have to natural logs. So in order to combine these two, I'm going to use the quotient rule. And if I do that I'll have natural log C squared. And this second value the square root of D becomes my denominator. So I have C squared divided by the square root of D. And notice now I have a single algorithm With a coefficient of one. So this becomes my final answer and you can see that this aligns with answer choice B. 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 - (1/3) ln y

Key Concepts

Properties of Logarithms

Natural Logarithm

Condensing Logarithmic Expressions

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