Find the areas between the curves y=2(log_2(x))/x and y=2(log_...

Question

Find the areas between the curves y=2(log_2(x))/x and y=2(log_4(x))/x and the x-axis from x=1 to x=e. What is the ratio of the larger area to the smaller?

Accepted Answer

Welcome back everyone. Consider the two curves y equals 4 log of base 5 of x divided by x and y equals 4 log base 25 of x divided by x along with the x axis on the interval. Of x between 1 and 5 inclusive, let the area enclosed by y equals 4 log base 5 of x divided by x, and the xx is from x equals 1 to x equals 5 b a 1, and the area enclosed by y equals 4 log base 25 of x divided by x, and the x x is over the same interval b a 2. What is the ratio of the larger area to the smaller? A 3 halves, B2, C 5 halves, and D 4? So for this problem, let's begin with the first area. Let's define a 1, and as the problem suggests, we want to define our integral from 1 to 5. Those are the limits of integration. And remember that the area between the curve and the x axis is the absolute value of our function, right? Considering the integral overall, so we're going to take. The integral from 1 to 5 of the absolute value of 4 log of base 5 of x divided by x. Followed by dx, notice that the limits of integration are from 1 to 5, meaning both x and the logarithm are non-negative, right? And therefore we can remove the absolute value. Additionally, what we're going to do is change the base of the logorithm so that log of base 5 of x can be written as ln x divided by ln of 5, right? So now we can show that our integral is the integral from 1 up to 5 of 4 multiplied by ln of x divided by ln of 5 multiplied by x and we can also factor out the constant. Specifically, we have 4 divided by. Ean of 5 followed by the integral from 1 to 5 of elan of x divided by x d x. So we can now show clearly that this is a one. And now let's consider a 2. Now for A2, the steps would be the same. We're integrating from 1 to 5, but the function differs. Again, since the limits of integration are from 1 to 5, we're not going to use the absolute value. We can just write 4 log. Of base 25 of x divided by XDX, right? And again we can show that log of base 25 of x is equal to ln x divided by ln of 25. Notice that ln of 25 is basically ln of 5 squared. So we can write it as ln of x divided by 2 ln of 5 using the properties of logarithms. So in this case, a 2 can be written as the integral from 1 up to 5 of 4, LN of x. Divided by 2 XLN or 5 D X, right? And now we can continue the equation and factor out the constant. So we are going to have 4 divided by 2 of 5. Integral from 1 to 5 of eon of X. Divided by XD X. And now we can go ahead and find the ratio. Notice that both. Areas contain the same integral and considering the constant. 4 divided by LN of 5 is greater than 4 divided by 2LN of 5, in particular, because our denominator is greater, this makes a 2 smaller. So we want to identify the ratio A1 divided by a 2. The ratio will allow us to cancel out each integral, right, so we can just consider our constants for a 1 we have 4 divided by lN of 5 and 4 a 2. We have 4 divided by. To a land of 5. Notice that we can cancel out 4 and Elena 5, which leaves us with. 1 divided by 1/2 or basically 2. So the correct answer for this problem corresponds to the answer choice B. Thank you for watching.

Find the areas between the curves y=2(log_2(x))/x and y=2(log_4(x))/x and the x-axis from x=1 to x=e. What is the ratio of the larger area to the smaller?

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