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10*3^((5t)/(4))=800 What is the solution of the equation? Round your answer, if necessary, to the nearest thousandth. t~~

10⋅35t4=800 10 \cdot 3^{\frac{5 t}{4}}=800 \newlineWhat is the solution of the equation?\newlineRound your answer, if necessary, to the nearest thousandth.\newlinet≈ t \approx

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Q. 10⋅35t4=800 10 \cdot 3^{\frac{5 t}{4}}=800 \newlineWhat is the solution of the equation?\newlineRound your answer, if necessary, to the nearest thousandth.\newlinet≈ t \approx
  1. Isolate exponential term: Isolate the exponential term.\newlineWe start by dividing both sides of the equation by 1010 to isolate the exponential term on one side.\newline10⋅3(5t)/4=80010 \cdot 3^{(5t)/4} = 800\newline3(5t)/4=800103^{(5t)/4} = \frac{800}{10}\newline3(5t)/4=803^{(5t)/4} = 80
  2. Apply logarithm: Apply the logarithm to both sides.\newlineTo solve for tt, we take the natural logarithm (ln⁡\ln) of both sides of the equation.\newlineln⁡(3(5t4))=ln⁡(80)\ln(3^{(\frac{5t}{4})}) = \ln(80)
  3. Use power property of logarithms: Use the power property of logarithms.\newlineThe power property of logarithms states that ln⁡(ab)=b⋅ln⁡(a)\ln(a^b) = b \cdot \ln(a). We apply this property to simplify the left side of the equation.\newline5t4⋅ln⁡(3)=ln⁡(80)\frac{5t}{4} \cdot \ln(3) = \ln(80)
  4. Isolate t: Isolate tt.\newlineTo solve for tt, we multiply both sides of the equation by 45\frac{4}{5} and divide by ln⁡(3)\ln(3).\newlinet=4⋅ln⁡(80)5⋅ln⁡(3)t = \frac{4 \cdot \ln(80)}{5 \cdot \ln(3)}
  5. Calculate value of t: Calculate the value of tt.\newlineNow we use a calculator to find the numerical value of tt.\newlinet=4⋅ln⁡(80)5⋅ln⁡(3)t = \frac{4 \cdot \ln(80)}{5 \cdot \ln(3)}\newlinet≈4⋅4.382026634675⋅1.09861228867t \approx \frac{4 \cdot 4.38202663467}{5 \cdot 1.09861228867}\newlinet≈17.528106538685.49306144335t \approx \frac{17.52810653868}{5.49306144335}\newlinet≈3.19153829085t \approx 3.19153829085
  6. Round answer: Round the answer to the nearest thousandth. t≈3.192t \approx 3.192

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