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Find the average value of the function f(x)=(8)/(x-1) from x=3 to x=7. Write your answer as the logarithm of a single number in simplest form. Answer: ln(◻)

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

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Q. Find the average value of the function f(x)=8x−1 f(x)=\frac{8}{x-1} from x=3 x=3 to x=7 x=7 . Write your answer as the logarithm of a single number in simplest form.\newlineAnswer: ln⁡(□) \ln (\square)
  1. Calculate Integral: To find the average value of a continuous function f(x)f(x) on the interval [a,b][a, b], we use the formula:\newlineAverage value = 1(b−a)∫abf(x) dx\frac{1}{(b-a)} \int_{a}^{b} f(x) \, dx\newlineHere, a=3a = 3 and b=7b = 7, so we need to calculate the integral of f(x)=8x−1f(x) = \frac{8}{x-1} from x=3x = 3 to x=7x = 7 and then divide by (b−a)(b-a).
  2. Evaluate Antiderivative: First, we calculate the integral of f(x)=8x−1f(x) = \frac{8}{x-1}. The antiderivative of 1x−1\frac{1}{x-1} is ln⁡∣x−1∣\ln|x-1|, so the antiderivative of 8x−1\frac{8}{x-1} is 8⋅ln⁡∣x−1∣8\cdot\ln|x-1|.
  3. Simplify Expression: Next, we evaluate the antiderivative from x=3x = 3 to x=7x = 7:∫378x−1 dx=[8ln⁡∣x−1∣]37\int_{3}^{7} \frac{8}{x-1} \, dx = [8\ln|x-1|]_{3}^{7}=8ln⁡∣7−1∣−8ln⁡∣3−1∣= 8\ln|7-1| - 8\ln|3-1|=8ln⁡(6)−8ln⁡(2)= 8\ln(6) - 8\ln(2)
  4. Calculate Average Value: Now we simplify the expression:\newline8⋅ln⁡(6)−8⋅ln⁡(2)=8⋅(ln⁡(6)−ln⁡(2))8\cdot\ln(6) - 8\cdot\ln(2) = 8\cdot(\ln(6) - \ln(2))\newlineUsing the properties of logarithms, ln⁡(a)−ln⁡(b)=ln⁡(ab)\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right), we get:\newline8⋅(ln⁡(62))=8⋅ln⁡(3)8\cdot(\ln(\frac{6}{2})) = 8\cdot\ln(3)
  5. Calculate Average Value: Now we simplify the expression:\newline8⋅ln⁡(6)−8⋅ln⁡(2)=8⋅(ln⁡(6)−ln⁡(2))8\cdot\ln(6) - 8\cdot\ln(2) = 8\cdot(\ln(6) - \ln(2))\newlineUsing the properties of logarithms, ln⁡(a)−ln⁡(b)=ln⁡(ab)\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right), we get:\newline8⋅(ln⁡(62))=8⋅ln⁡(3)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)(b-a), which is (7−3)(7-3):\newlineAverage value = 1(7−3)⋅8⋅ln⁡(3)\frac{1}{(7-3)} \cdot 8\cdot\ln(3)\newline= 2⋅ln⁡(3)2\cdot\ln(3)\newline= ln⁡(32)\ln(3^2)\newline= ln⁡(9)\ln(9)

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