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.

log x + log(x^2 - 1) - log 7 - log(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 into a single logarithms with a coefficient of one and we want to try to evaluate without a calculator and the expression is natural log y plus natural log y squared minus nine minus natural log three minus natural log y plus three. So in this case you can see that we have logarithms that are being added and also subtracted from one another. And when we're trying to combine these, we want to remember the logarithmic properties. So recall that if we have two logarithms, say natural log A plus natural log B, we can combine those when we have multiplication. Using the product rule where natural log A plus natural log B becomes natural log A times B. That is known as the product rule. And if we have a difference between two logs, say natural log a minus natural log B. That is the quotient rule and that becomes natural log A divided by B. When we have addition, it's the product rule. And then when we have subtraction it's the quotient rule. And essentially addition means multiplication and subtraction means division. So if we look at our equation, you can see that we have the first two logarithms are being added together. So that means natural log y plus natural log y squared minus nine becomes natural log y times y squared minus nine. And I still have these last two logarithms minus. Natural log three minus natural log y plus three. And like we said subtraction is division. So if I combine these lost too into my single algorithm, I'll be loved with natural log y times y squared minus nine divided by three times Y plus three. So notice how The two logarithms that had subtraction in front of them appeared in the denominator of my single algorithm. And from here we can simplify further. If we look at why squared minus nine that is a difference of squares. So I can rewrite this as Y plus three times y minus three. So if I go back to my logo rhythm and I write Y squared minus nine as Y plus three times y minus three my numerator becomes Y times Y plus three times y minus three. And that's still being divided by three times Y plus three. But now notice how I can cancel out Y plus three since it's both in the numerator and the denominator And that cancels out to one. So my final algorithm is natural log Y times Y minus three, divided by three. So that becomes my single algorithm and you can see that we have a coefficient of one since there's no number in front of this log. 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

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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. log x + log(x^2 - 1) - log 7 - log(x + 1)