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

Full solution

Q. A culture of bacteria starts with

50

bacteria and increases exponentially.

\newline

The relationship between

B

, the number of bacteria in the culture, and

d

, the elapsed time, in days, is modeled by the following equation.

\newline

B=50 \cdot 10^{\frac{d}{2}}

\newline

In how many days will the number of bacteria in the culture reach

800

000

\newline

Give an exact answer expressed as a base-ten logarithm.

\newline

days

Given exponential growth model: We are given the exponential growth model for the bacteria culture: $\newline$ $B = 50 \times 10^{(d)/(2)}$ $\newline$ We want to find the value of $d$ when $B$ is $800,000$ . $\newline$ First, we set up the equation with $B = 800,000$ : $\newline$ $800,000 = 50 \times 10^{(d)/(2)}$
Set up equation: To solve for $d$ , we first divide both sides of the equation by $50$ to isolate the exponential term: $\newline$ $\frac{800,000}{50} = 10^{\left(\frac{d}{2}\right)}$
Isolate exponential term: Perform the division on the left side of the equation: $16,000 = 10^{(d)/(2)}$
Take base-ten logarithm: Now, we take the base-ten logarithm of both sides to solve for $d/2$ : $\log(16,000) = \log(10^{(\frac{d}{2})})$
Simplify right side: Using the property of logarithms that $\log(a^b) = b\cdot\log(a)$ , we can simplify the right side of the equation: $\newline$ $\log(16,000) = \left(\frac{d}{2}\right) \cdot \log(10)$ $\newline$ Since $\log(10)$ is $1$ , this simplifies to: $\newline$ $\log(16,000) = \frac{d}{2}$
Multiply both sides: Now, we multiply both sides by $2$ to solve for $d$ : $\newline$ $2 \times \log(16,000) = d$
Calculate value of d: We can now calculate the value of $d$ using a calculator: $d = 2 \times \log(16,000)$
Final answer: The exact answer expressed as a base-ten logarithm is: $\newline$ d = $2 \times \log(16,000)$ $\newline$ This is the final answer in the form requested.