logarithm help?

Question

I'm confused as how to apply log laws when it comes to solving problems such as this. Where would everything be appropriate.
Step by step help would be greatly appreciated. It would help to answer other answers aswell.

Bacteria is growing according to formula M=M(subscript o)x10^np, where M is the number present after p weeks.
-State M(subscript 0) if the number is initially 400
-Find n if the number present is doubled after 3 weeks.
-Find how long it takes to the nearest day, to triple the original number.

Answer 1

I think this is what you are looking for.

M = Mo x 10^np
Mo = 400
M = Mo x 2 = 800 after three weeks ( p is 3).

M = Mo x 10^np
800 = 400 x 10^n3
800/400 = 10^n3
2 = 10^3n Take logs.
log 2 = log10^3n
0∙301 029 995 = 3n(1)
0∙301 029 995 / 3 = n
n = 0∙100 343 331 Take anti-log.

To find p:
M = Mo x 10^np
1200 = 400 x 10^0∙100 343 331p
1200 / 400 = 10^0∙100 343 331p
3 = 10^0∙100 343 331p Take logs.
Log 3 = Log 0∙100 343 331p
0∙477 121 254 = 0∙100 343 331p (1)
0∙477 121 254 = 0∙100 343 331p
0∙477 121 254 / 0∙100 343 331 = p
p = 4∙754 887 537
p = 4∙754 887 537 Weeks.

Days = p x 7
Days = 4∙754 887 537 x 7
Days = 33∙284 2212 76
Days ≈ 33

Answer 2

Let's use A instead of M sub 0. Your growth formula is M = A • 10^(np).

-- If "the number is initially 400", then A (M sub 0) is 400

-- If A doubles in 3 weeks, M = 2A, so we have
2A = A • 10^(3n)
2 = 10^(3n) ......... and using base 10 logs,
log 2 = log [ 10^(3n)] = 3n
(log 2)/3 = n = 0.10034

To triple A using n = 0.10034, then M = 3A and we get
3A = A • 10^( 0.10034p)
3 = 10^(0.10034p)
log 3 = 0.10034p
p = log 3/0.10034 = 4.755 weeks

An exact answer requires simplifying 10^[(log 2)/3] = [10^(log 2)]^(1/3) = 2^(1/3) = cube root of 2.
So you have 3 = (cube root of 2)^p
log 3 = p ( log of cube root of 2)
p = log 3 / (log of cube root of 2)

Answer 3

M0 is 400 since it's the initial population.
M = M0 10^(np), so log(M/M0) = np. If we know that p=3 (weeks), and if we know that the number doubled (M / M0 = 2), then log(2) = 3n, and n = log(2)/3 or 0.100343332.

For the last question, if M/M0 = 3, and n is the value above, p = log(M/M0) / n, so p = log(3) / 0.100343332 or 4.75489 weeks. This is 4 weeks and 7 x 0.75489 ( 5.28 ) days.