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.

(1/2)(log5 x + log5 y) - 2 log5 (x + 1)

Accepted Answer

Hello. Everyone in this video. We're going to be looking at this practice problem where we want to condense the equation one third times log base three A plus log base three B minus eight Log Base three A Plus B. And we want to condense the equation into a single algorithm with a coefficient of one. And we're going to try to condense it or evaluate it without using a calculator in this equation. You can see that we have two parts, we have this one third times log base three A. Plus log base three B. And then we have the second part where we have the minus eight log base three A plus B. And in order to evaluate these without using a calculator, it's important to remember the properties for condensing logarithms. So one of the properties to remember is known as the product rule. And the product rule states that when you have a logarithms say we have log base B M. Plus log base B N. We can condense that into a single algorithm by saying that log base B M plus log base B N is equal to log base B. M. Times. And so notice how the addition turned into a multiplication. And we went in with two logarithms and we came out with one and we can apply this same sort of rule with subtraction using the quotient rule. So now we're going to go in with log base B M minus log base B N. Notice how the logarithms have the same base and now they're being subtracted. We can represent this as a single log using division where now we write this as log base B. The first algorithm M over n. Notice how again we went in with two logarithms and we came out with one. And then the last rule is the power rule where if there is a coefficient in front of the log say P log base B, we can rewrite this P as an exponent. So now we only have log base B. M to the power of P. So notice how now the coefficient in front of the log is simply one. So we're going to use all of these rules to simplify our equation. And if we start with this first part where we have 1/3 log, base three A. Plus log base three B. Let's look at this parentheses first and you can see that we have an addition between logs. So I'm going to use the product rule to combine this into one log. So this will become log base three A times B And I still have the 1/3. And now I can apply the power rule to get rid of the 1/3. Where I can now write this as log base three A times b to the power of 1/3. And then I can also deal with the second part where I have eight Log Base three A Plus B. And I can move this eight into the logarithms. So I can rewrite this as log base three A plus B to the eighth. Power. And if I combine this first part, log base three A. B. To the one third minus because we have this minus still minus the second part which is log base three A plus B to the eighth. You can see that we can use the quotient rule to combine these last two logarithms. So this becomes log base three A. Times B to the one third, Divided by a Plus B to the 8th. And this becomes our final answer. And because we have the one third, you can also write this as log base three A. B. And we're taking the third root of it And we still have the denominator a plus B to the 8th. So I just rewrote it Because the power of 1/3 is the same thing as taking the 3rd route. So you can see that this answer aligns with answer choice D. So hopefully this video hope can 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. (1/2)(log5 x + log5 y) - 2 log5 (x + 1)

Key Concepts

Properties of Logarithms

Condensing Logarithmic Expressions

Evaluating Logarithmic Expressions

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