Find the average value of the function f(x)=(8)/(x-3) from x=5 to x=7. Write your answer as the logarithm of a single number in simplest form. Answer: ln(◻)

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Find the average value of the function  f(x)=(8)/(x-3) from  x=5 to  x=7. Write your answer as the logarithm of a single number in simplest form. Answer:  ln(◻)

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Set Up Integral: To find the average value of a continuous function f(x) on the interval [a, b], we use the formula: Average value = (1/(b-a)) * ∫ from a to b of f(x) dx Here, a = 5 and b = 7, so we need to integrate f(x) = (8)/(x-3) from x = 5 to x = 7 and then multiply by 1/(7-5).Solve Integral: First, let's set up the integral: Average value = (1/(7-5)) * ∫ from 5 to 7 of (8)/(x-3) dx This simplifies to: Average value = (1/2) * ∫ from 5 to 7 of (8)/(x-3) dxEvaluate Integral: To integrate (8)/(x-3), we recognize this as a simple logarithmic integration problem. The integral of 1/u du is ln|u|, so the integral of (8)/(x-3) dx is 8ln|x-3|. Now we need to evaluate this from x = 5 to x = 7.Simplify Expression: Evaluating the integral, we get: (1/2) * [8ln|7-3| - 8ln|5-3|] This simplifies to: (1/2) * [8ln(4) - 8ln(2)]Apply Logarithm Properties: We can simplify the expression using logarithm properties. The difference of logarithms is the logarithm of the quotient: (1/2) * 8ln(4/2) Since 4/2 is 2, this further simplifies to: (1/2) * 8ln(2)Final Result: Multiplying through by (1/2) * 8 gives us: 4ln(2) This is the average value of the function f(x) from x = 5 to x = 7, expressed as the logarithm of a single number in simplest form.

Find the average value of the function $f(x)=\frac{8}{x-3}$ from $x=5$ to $x=7$ . Write your answer as the logarithm of a single number in simplest form. $\newline$ Answer: $\ln (\square)$

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Find the average value of the function $f(x)=\frac{8}{x-3}$ from $x=5$ to $x=7$ . Write your answer as the logarithm of a single number in simplest form. $\newline$ Answer: $\ln (\square)$