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-7*3^(0.25 x)=-10 Which of the following is the solution of the equation? Choose 1 answer: (A) x=(4log_(3)(10))/(log_(3)(7)) (B) x=4log_(3)((10)/(7)) (C) x=(4log_(3)(7))/(log_(3)10) (D) x=4log_((10)/(7))(3)

−7⋅30.25x=−10 -7 \cdot 3^{0.25 x}=-10 \newlineWhich of the following is the solution of the equation?\newlineChoose 11 answer:\newline(A) x=4log⁡3(10)log⁡3(7) x=\frac{4 \log _{3}(10)}{\log _{3}(7)} \newline(B) x=4log⁡3(107) x=4 \log _{3}\left(\frac{10}{7}\right) \newline(C) x=4log⁡3(7)log⁡310 x=\frac{4 \log _{3}(7)}{\log _{3} 10} \newline(D) x=4log⁡107(3) x=4 \log _{\frac{10}{7}}(3)

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Q. −7⋅30.25x=−10 -7 \cdot 3^{0.25 x}=-10 \newlineWhich of the following is the solution of the equation?\newlineChoose 11 answer:\newline(A) x=4log⁡3(10)log⁡3(7) x=\frac{4 \log _{3}(10)}{\log _{3}(7)} \newline(B) x=4log⁡3(107) x=4 \log _{3}\left(\frac{10}{7}\right) \newline(C) x=4log⁡3(7)log⁡310 x=\frac{4 \log _{3}(7)}{\log _{3} 10} \newline(D) x=4log⁡107(3) x=4 \log _{\frac{10}{7}}(3)
  1. Isolate exponential term: Isolate the exponential term.\newlineWe start by isolating the exponential term on one side of the equation.\newline−7⋅3(0.25x)=−10-7 \cdot 3^{(0.25 x)} = -10\newlineDivide both sides by −7-7 to isolate the exponential term.\newline3(0.25x)=−10−73^{(0.25 x)} = \frac{-10}{-7}\newline3(0.25x)=1073^{(0.25 x)} = \frac{10}{7}
  2. Take logarithm of both sides: Take the logarithm of both sides.\newlineTo solve for xx, we take the logarithm of both sides. We can use any logarithm base, but it's convenient to use base 33 because of the base of the exponential term.\newlinelog⁡3(30.25x)=log⁡3(107)\log_3(3^{0.25 x}) = \log_3(\frac{10}{7})
  3. Apply power rule of logarithms: Apply the power rule of logarithms.\newlineThe power rule of logarithms states that log⁡b(ac)=clog⁡b(a)\log_b(a^c) = c\log_b(a). We apply this rule to the left side of the equation.\newline0.25x⋅log⁡3(3)=log⁡3(107)0.25 x \cdot \log_3(3) = \log_3(\frac{10}{7})\newlineSince log⁡3(3)=1\log_3(3) = 1, the equation simplifies to:\newline0.25x=log⁡3(107)0.25 x = \log_3(\frac{10}{7})
  4. Solve for x: Solve for x.\newlineTo solve for x, we divide both sides by 0.250.25.\newlinex=log⁡3(107)0.25x = \frac{\log_3(\frac{10}{7})}{0.25}\newlinex=4×log⁡3(107)x = 4 \times \log_3(\frac{10}{7})
  5. Match solution with choices: Match the solution with the given choices.\newlineThe solution we found is x=4⋅log⁡3(107)x = 4 \cdot \log_3(\frac{10}{7}), which matches choice (B).

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