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A culture of bacteria starts with 50 bacteria and increases exponentially. The relationship between B, the number of bacteria in the culture, and d, the elapsed time, in days, is modeled by the following equation. B=50*10^((d)/(2)) In how many days will the number of bacteria in the culture reach 800,000 ? Give an exact answer expressed as a base-ten logarithm. days

A culture of bacteria starts with 5050 bacteria and increases exponentially.\newlineThe relationship between B B , the number of bacteria in the culture, and d d , the elapsed time, in days, is modeled by the following equation.\newlineB=50⋅10d2 B=50 \cdot 10^{\frac{d}{2}} \newlineIn how many days will the number of bacteria in the culture reach 800800,000000 ?\newlineGive an exact answer expressed as a base-ten logarithm.\newlinedays

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Q. A culture of bacteria starts with 5050 bacteria and increases exponentially.\newlineThe relationship between B B , the number of bacteria in the culture, and d d , the elapsed time, in days, is modeled by the following equation.\newlineB=50⋅10d2 B=50 \cdot 10^{\frac{d}{2}} \newlineIn how many days will the number of bacteria in the culture reach 800800,000000 ?\newlineGive an exact answer expressed as a base-ten logarithm.\newlinedays
  1. Given exponential growth model: We are given the exponential growth model for the bacteria culture:\newlineB=50×10(d)/(2)B = 50 \times 10^{(d)/(2)}\newlineWe want to find the value of dd when BB is 800,000800,000.\newlineFirst, we set up the equation with B=800,000B = 800,000:\newline800,000=50×10(d)/(2)800,000 = 50 \times 10^{(d)/(2)}
  2. Set up equation: To solve for dd, we first divide both sides of the equation by 5050 to isolate the exponential term:\newline800,00050=10(d2)\frac{800,000}{50} = 10^{\left(\frac{d}{2}\right)}
  3. Isolate exponential term: Perform the division on the left side of the equation: 16,000=10(d)/(2)16,000 = 10^{(d)/(2)}
  4. Take base-ten logarithm: Now, we take the base-ten logarithm of both sides to solve for d/2d/2:log⁡(16,000)=log⁡(10(d2))\log(16,000) = \log(10^{(\frac{d}{2})})
  5. Simplify right side: Using the property of logarithms that log⁡(ab)=b⋅log⁡(a)\log(a^b) = b\cdot\log(a), we can simplify the right side of the equation:\newlinelog⁡(16,000)=(d2)⋅log⁡(10)\log(16,000) = \left(\frac{d}{2}\right) \cdot \log(10)\newlineSince log⁡(10)\log(10) is 11, this simplifies to:\newlinelog⁡(16,000)=d2\log(16,000) = \frac{d}{2}
  6. Multiply both sides: Now, we multiply both sides by 22 to solve for dd: \newline2×log⁡(16,000)=d2 \times \log(16,000) = d
  7. Calculate value of d: We can now calculate the value of dd using a calculator: d=2×log⁡(16,000)d = 2 \times \log(16,000)
  8. Final answer: The exact answer expressed as a base-ten logarithm is:\newlined = 2×log⁡(16,000)2 \times \log(16,000)\newlineThis is the final answer in the form requested.

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