How long will it take unitl there are 1 000 000 backteria?

[wind]

Hi can someone check over my work?

There are initially 5000 backteria in a culture. The number of bacteria doubles every 2h, so the number of bacteria after t hours will be

\(\displaystyle \L \ N= 5000 (2)^{\frac{t}{2}}\)

How long will it take unitl there are 1 000 000 backteria present?

tyring to solve for t

\(\displaystyle \L \ 1 ,000, 000= 5000 (2)^{\frac{t}{2}}\)
.
\(\displaystyle \L \ log 1, 000 ,000= log 5,000 (2)^{\frac{t}{2}}\)
.
\(\displaystyle \L \ log 1 ,000 ,000= \frac{t}{2} log 5,000 (2)\)
.
\(\displaystyle \L \frac {log 1 ,000 ,000}{log10,000}= \frac{t}{2}\)
.
\(\displaystyle \L \1.5= \frac{t}{2}\)
.
\(\displaystyle \L \3= {t}\)

but this is not the answer in the back of the book :?

Thanks

 

[galactus]

You should've divided through by 5000 first.

\(\displaystyle \L\\5000\cdot{2^{\frac{t}{2}}}=1,000,000\)

\(\displaystyle \L\\2^{\frac{t}{2}}=200\)

\(\displaystyle \frac{t}{2}ln(2)=ln(200}\)

\(\displaystyle \L\\\frac{t}{2}=\frac{ln(200)}{ln(2)}\)

\(\displaystyle \L\\t=\frac{2ln(200)}{ln(2)}\approx{15.29}\)

 

[wind]

Thanks galactus