In Exercises 1–40, use properties of logarithms to expand each lo...

Question

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.

log [(10x^2∛(1 - x))/(7(x + 1)^2)]

Accepted Answer

Hello. Today we are going to be expanding the given logarithms. So the logarithms given to us is the logo rhythm of 100 times Y. To the fourth times the square root of two minus Y. Over three times Y plus eight cubed. Now this looks like a big mess. But the first thing you can go ahead and do is utilize the division property for logarithms, which states that if you have the logarithms of A over B, this can be rewritten as the logo rhythm of A minus the logarithms of B. Now here are A is going to be this entire numerator and RB is going to be the entire denominator. And what this gives us is the logo rhythm of times Y to the fourth times the square root of two minus Y minus the logo rhythm of three times Y plus eight. Cute. Okay, now notice that inside both of these logarithms are quantities that are being multiplied by each other. So if we utilize the multiplication property for logarithms, which states that if you have the logo rhythm of A. Times B. This can be rewritten as the logo rhythm of A. Plus the lagoon rhythm of B. And we're going to be applying this property once to the right hand, logo rhythm and twice to the left hand logo rhythm. So applying it to the first one specifically to 100 Y. To the fourth times the square root of two minus Y. Is going to give us the logarithms of 100 Y. To the fourth plus the logarithms of the square root of two minus Y. And that's going to be minus ng the logarithms of three plus the logarithms of Y plus eight cubed. Okay now what we want to go ahead and do is utilize the multiplication property one more time on the logarithms of 100 Y to the fourth because 100 y to the fourth is a product. So what this gives us is the logarithms of plus the logarithms of y to the fourth plus the logarithms of the square root of two minus y minus the logarithms of three plus the logarithms of Y plus eight cubed. Okay and I'm gonna go ahead and rewrite a couple of things because we want to go ahead and get exponents instead of her logarithms. 100 can be rewritten as 10 squared. So the first logarithms can be rewritten as the logo rhythm of 10 squared and the square root of two minus Y is the same thing as the quantity of two minus y raised to the power of a half. So we can go ahead and rewrite this argument as the quantity of two minus y, raising the power of one half. Now what we can go ahead and do is utilize the exponential rule of logarithms and that states that if you have the logo rhythm of A to the power of B. This can be rewritten as B times the logarithms of A. And we're going to be applying this exponential rule to the 1st, 2nd, 3rd and 5th logarithms and this is going to give us two times the logarithm of plus four times the logarithm of why, plus one half times the logarithms of two minus y minus the logarithm of three plus three times the logarithms of Y plus eight. Now there is one more simplification that we can do recall that. The general logarithms has an automatic base of 10 and if you have a logarithms whose base matches its argument, you can rewrite that logarithms as one. So what we have here is two times one and two times one is just going to give us two. So simplifying this one more time is going to give us two plus four times the logarithms of why? Plus one half times the logarithms of two minus y minus the logarithm of three plus three times the logo rhythm of Y plus eight. And this is gonna be the complete expansion of our original logarithms. And that means that the answer to this problem is going to be a. So I hope this video helps you in understanding how to, how to expand a given logarithms and I'll go ahead and see you all in the next video

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In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log [(10x^2∛(1 - x))/(7(x + 1)^2)]