Use properties of logarithms to condense each logarithmic expr...

Question

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. $\frac{1}{3} \left[ 2 \ln(x + 5) - \ln x - \ln (x^2 - 4) \right]$

Accepted Answer

Hello everyone in this video. We're going to be looking at this practice problem where we're given the expression one half times three times natural log a plus 16 minus natural log three a minus Natural log a squared minus 64. And we want to try to combine that expression into a single algorithm with a coefficient of one. And also we want to try to evaluate without using a calculator, which means that we're going to be relying on logarithmic properties. So in this case we're trying to get rid of these coefficients in the front and recall that the power rule for logarithms could be helpful. Say we have logarithms or a natural log with a coefficient P in front. So say I have P times natural log A. I can rewrite this with the P as an exponent where p times natural log A becomes natural log A to the power of P. So notice how now once I apply the power rule, I have a coefficient of one in front of this. Natural log. And we also have some subtraction here. Because we're trying to get down to a single algorithm. We can also recall the quotient rule and the quotient rule states that when you have two natural logs a natural log a minus natural log B. We can combine them into a fraction as natural log a divided by B. So notice how we go in with two natural logs but we come out with one. So that's going to be helpful when we're trying to combine the three natural logs that we have in the expression into one. So let's use these rules to start to simplify this expression. So I'm gonna start with this first chunk where we have three times natural log a plus 16. So from here you can see that we want to get rid of that three. So I'm gonna go ahead and apply the power rule. So this becomes Natural log a plus 16 And a plus 16 to the power of three. So notice how I applied the power of three to the whole expression a plus 16 and not just to the variable a. So now I've simplified that and I still have the one half in front of that entire expression, but I now have one half times natural log a plus 16 to the third minus natural log three a minus natural log a squared minus 64. So in this case now we have these minus natural logs that we can deal with using the quotient rule. So if I do that, I can rewrite this as a single logarithmic. Where I have the natural log a plus 16 to the third is in my numerator and in my denominator I'll have this first subtraction where I have +38 and then this second subtraction where I have a squared minus 64. So my denominator is three a times a squared -64. So from here you can see that I now have only one natural log but I still don't have a coefficient of one. I have this coefficient one half. So we can go ahead and apply the power rule again. So this becomes natural log a plus 16 to the third, divided by three a times a squared minus 64. And I need to apply the power of one half to this whole expression. So I'm going to put that in parentheses and Apply the 1/ and recall that anything to the power of one half is taking the square root of that expression. So I can rewrite this as natural log square root of a plus 16 to the third, divided by three A times a squared minus 64. And you can see that I now have a coefficient of one and I have a single natural log. So this becomes my final answer. And notice how we evaluated that. Using only the logarithmic properties and no calculator. And just by following quotient rule and power rule, we are able to condense the expression into the single log. And if you look at the answer choices, answer choice A aligns with what we got. So hopefully this video helped and see you next time

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. $\frac{1}{3} \left[ 2 \ln(x + 5) - \ln x - \ln (x^2 - 4) \right]$

Key Concepts

Properties of Logarithms

Natural Logarithm (ln)

Simplifying Algebraic Expressions Inside Logarithms

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