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.4 ln (x + 6) - 3 ln x

Accepted Answer

Hello, everyone in this video. We're going to be looking at this practice problem where we want to condense the equation eight times natural log y plus one minus five times natural log y into a single algorithm with a coefficient of one and we want to do that without a calculator. So because we're doing without a calculator and we're working with logarithms, we're going to be relying on the logarithmic properties. So recall that we have different logarithmic properties that can help us in condensing logarithms. So in this case we have two logarithms and we only have want one, there's a subtraction between the two logarithms and there's also coefficients in front of the logarithms where we want a coefficient of one. So when there's subtraction between logarithms, say we have natural log a minus natural log B. We can condense this into one logarithms. Using the quotient rule and using the quotient rule logarithms. Natural log a minus natural log B becomes natural log A divided by B. So notice how the value of the second natural log became the denominator. And there's also another rule known as the power rule where we can move the coefficients of a natural log into the natural log. So say I have a times natural log B. I can move the A coefficient into the natural log by making it an exponent of B. So a times natural log B becomes natural log B to the power of A. So we're going to use both of these rules to condense our equation. So first I'm going to work with the power rule because as I said before these both have coefficients. So this first natural log has coefficient A and the second one has coefficient five. So if I use the power rule, going to rewrite this as natural log Y plus one to the power of eight and notice how I applied the power to the whole expression Y plus one and not just the Y. Because I need to apply it to the entire value of the natural log. And then I'm going to do the same to the second portion where I have five as a coefficient. So I'm going to make it the exponents of why? So now I have minus natural log Y to the fifth power. So notice how I no longer have any more coefficients, but I still have two natural logs and I only want one logarithms. So I'm going to combine these using the quotient rule where my first value is Y plus one to the eighth and my second value is Y to the fifth. So I'll write this as one natural log as natural log Y plus one to the eighth, divided by the value of the second algorithm, which is why to the fifth. So now you can see that 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 C. So hopefully this video hoped 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.4 ln (x + 6) - 3 ln x

Key Concepts

Properties of Logarithms

Natural Logarithm

Condensing Logarithmic Expressions

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