The number

N of bacteria in a culture at time t
(in hours) grows exponentially according to the function N(t) = 1000e^0.01t

When will the number of bacteria reach 1700?
I though you would do 1700=1,000^e0.01t and solve for t but i keep getting the wrong answer can someone please help me and show all work thank you

1.7 = e^0.01t

ln(1.7) = 0.01 t

100 ln(1.7) = t

To find the time at which the number of bacteria reaches 1700, we can set up the equation N(t) = 1700 and solve for t. Here's the step-by-step process:

1. Start with the exponential growth equation: N(t) = 1000e^(0.01t)
2. Set N(t) equal to 1700: 1700 = 1000e^(0.01t)
3. Divide both sides of the equation by 1000: 1.7 = e^(0.01t)
4. Take the natural logarithm (ln) of both sides to get rid of the exponential function: ln(1.7) = ln(e^(0.01t))
5. Simplify the right side using the property ln(e^x) = x: ln(1.7) = 0.01t
6. Divide both sides of the equation by 0.01: ln(1.7)/0.01 = t
7. Use a calculator to evaluate ln(1.7) approximately: ln(1.7) ≈ 0.53063
8. Divide 0.53063 by 0.01 to find t: t ≈ 53.063

So, the number of bacteria will reach 1700 after approximately 53.063 hours.

To find the time t when the number of bacteria reaches 1700, you need to solve the equation N(t) = 1700.

The given exponential function is:
N(t) = 1000e^(0.01t)

Substitute N(t) with 1700:
1700 = 1000e^(0.01t)

To solve this equation for t, we need to isolate the exponential term. We'll start by dividing both sides of the equation by 1000:
1700/1000 = e^(0.01t)

Simplifying the left side:
1.7 = e^(0.01t)

To remove the exponential, we'll take the natural logarithm (ln) of both sides:
ln(1.7) = ln(e^(0.01t))

Using the property of logarithms (ln(e^x) = x), the equation becomes:
ln(1.7) = 0.01t

Now, divide both sides by 0.01 to solve for t:
t = ln(1.7) / 0.01

Using a calculator, find the natural logarithm of 1.7: ln(1.7) ≈ 0.5306

Substituting this value into the equation:
t = 0.5306 / 0.01

Finally, calculate:
t ≈ 53.06

Therefore, the number of bacteria will reach 1700 after approximately 53.06 hours.