1. ## logarithms question

can someone please explain how to solve these questions?

The number of mold spores in a petri dish increases by a factor of 10 every week. If there are initially 40 spores in the dish, how long with it take for there to be 2000 spores?

and

The astronomer johannes kepler determined that the time, D, in days, for a planet to revolve around the sun is related to the planet's average distance from the sun, k, in millions of kilometres. This relation is defined by the equation log D = (3/2)logk - 0.7. Estimate the period of revolution of uranus about the sun, given its distance from the sun. Uranus is 2854 million kilometres away from the sun

2. ## Logs

Hello mzto
Originally Posted by mzto
can someone please explain how to solve these questions?

The number of mold spores in a petri dish increases by a factor of 10 every week. If there are initially 40 spores in the dish, how long with it take for there to be 2000 spores?
At time $\displaystyle t = 0$, there are 40 spores.

After 1 week, $\displaystyle t=1$, there are $\displaystyle 40\times 10$ spores

After 2 weeks, $\displaystyle t=2$, there are $\displaystyle 40\times 10\times 10 = 40\times 10^2$ spores

... and so on.

After $\displaystyle t$ weeks, there are $\displaystyle 40\times 10^t$ spores

So, if there are 2000 spores at time $\displaystyle t$, $\displaystyle 40\times 10^t = 2000$

$\displaystyle \Rightarrow 10^t = \frac{2000}{40}= 50$

$\displaystyle \Rightarrow t=\log(50)$, because the log (to base 10) of a number is the power to which you must raise 10 to get the number. (Do you understand this sentence? It's vital that you do, because you'll need it again in the second question - see below!)

$\displaystyle \Rightarrow t =1.699$ weeks, or 11 days 21.4 hours.

The astronomer johannes kepler determined that the time, D, in days, for a planet to revolve around the sun is related to the planet's average distance from the sun, k, in millions of kilometres. This relation is defined by the equation log D = (3/2)logk - 0.7. Estimate the period of revolution of uranus about the sun, given its distance from the sun. Uranus is 2854 million kilometres away from the sun
We need to find D when $\displaystyle k = 2854$.

So $\displaystyle \log D = \tfrac32\log(2854) - 0.7$

$\displaystyle = 4.48318...$

So $\displaystyle 4.48318$ is the power to which we must raise $\displaystyle 10$ in order to get $\displaystyle D$ (see question 1). In other words:

$\displaystyle D = 10^{4.48318} = 30421$ days

(Note that you can probably do this calculation on your calculator in two ways: one is to enter 4.48318 and then press INV and LOG, and the other is to use the x^y button, entering 10 for x and 4.48318 for y.)