Find the average value of the function $f(x)=\frac{8}{x-1}$ from $x=3$ to $x=7$ . Write your answer as the logarithm of a single number in simplest form. $\newline$ Answer: $\ln (\square)$

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Q. Find the average value of the function

f(x)=\frac{8}{x-1}

from

x=3

x=7

. Write your answer as the logarithm of a single number in simplest form.

\newline

Answer:

\ln (\square)

Calculate Integral: To find the average value of a continuous function $f(x)$ on the interval $[a, b]$ , we use the formula: $\newline$ Average value = $\frac{1}{(b-a)} \int_{a}^{b} f(x) \, dx$ $\newline$ Here, $a = 3$ and $b = 7$ , so we need to calculate the integral of $f(x) = \frac{8}{x-1}$ from $x = 3$ to $x = 7$ and then divide by $(b-a)$ .
Evaluate Antiderivative: First, we calculate the integral of $f(x) = \frac{8}{x-1}$ . The antiderivative of $\frac{1}{x-1}$ is $\ln|x-1|$ , so the antiderivative of $\frac{8}{x-1}$ is $8\cdot\ln|x-1|$ .
Simplify Expression: Next, we evaluate the antiderivative from $x = 3$ to $x = 7$ : $\int_{3}^{7} \frac{8}{x-1} \, dx = [8\ln|x-1|]_{3}^{7}$ $= 8\ln|7-1| - 8\ln|3-1|$ $= 8\ln(6) - 8\ln(2)$
Calculate Average Value: Now we simplify the expression: $\newline$ $8\cdot\ln(6) - 8\cdot\ln(2) = 8\cdot(\ln(6) - \ln(2))$ $\newline$ Using the properties of logarithms, $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$ , we get: $\newline$ $8\cdot(\ln(\frac{6}{2})) = 8\cdot\ln(3)$
Calculate Average Value: Now we simplify the expression: $\newline$ $8\cdot\ln(6) - 8\cdot\ln(2) = 8\cdot(\ln(6) - \ln(2))$ $\newline$ Using the properties of logarithms, $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$ , we get: $\newline$ $8\cdot(\ln(\frac{6}{2})) = 8\cdot\ln(3)$ Finally, we calculate the average value by dividing the result of the integral by $(b-a)$ , which is $(7-3)$ : $\newline$ Average value = $\frac{1}{(7-3)} \cdot 8\cdot\ln(3)$ $\newline$ = $2\cdot\ln(3)$ $\newline$ = $\ln(3^2)$ $\newline$ = $\ln(9)$