1. ## Logarithms

We have a function such that $\displaystyle f(t) = 10 \cdot 2^{kt}$ and $\displaystyle f(\tfrac{1}{2}) = 3$. Find $\displaystyle k$.

$\displaystyle f(\tfrac{1}{2}) = 3$, therefore:

$\displaystyle 3 = 10 \cdot 2^{\tfrac{k}{2}}$

Taking the log to the base 2 on both sides:

$\displaystyle log_2 3 = log_2 10 + \frac{k}{2}$

Now, I know that $\displaystyle log_a xy = log_a x + log_a y$, but this must mean that we had:

$\displaystyle log_2 3 = log_2 10 + log_2 2^{\tfrac{k}{2}}$

Can someone show me why (proof) that $\displaystyle log_2 2^{\tfrac{k}{2}} = \frac{k}{2}$?

And is it just me, or do those log renders look a bit squashed? The logs and the fractional exponent. Is there a better way to render these?

2. $\displaystyle 3 = 10 \cdot 2^{\tfrac{k}{2}}$

Divide both sides by ten ... it should now look like this

$\displaystyle$
$\displaystyle \frac{3}{10}= 2^{kt}$

Now

$\displaystyle$
$\displaystyle log 3 - log 10 = \frac{k} {2} log 2$

multiply both sides by 2

$\displaystyle 2(log 3 - log 10) = k log 2$

Divide both sides by log 2 you have now isolated k

3. Thanks for the solution, but I am interested in finding out why first:

$\displaystyle log_2 2^{\tfrac{k}{2}} = \frac{k}{2}$

I'm not familiar with the proof or general rule for why this is.

4. Originally Posted by rowe
Thanks for the solution, but I am interested in finding out why first:

$\displaystyle log_2 2^{\tfrac{k}{2}} = \frac{k}{2}$

I'm not familiar with the proof or general rule for why this is.
If you're happy with $\displaystyle log_c(ab) = log_c(a)+log_c(b)$ then that can be used (although I'm sure someone can make it more elegant

$\displaystyle x^n = x \cdot x \cdot x ...$ up to n times

$\displaystyle log_2(x \cdot x \cdot x ... \cdot x)$

$\displaystyle = log_2(x) + log_2(x) + log_2(x) + ... + log_2(x)$

Factor out like terms (and there are n of them)

$\displaystyle n(log_2(x))$

Wikipedia defines it in terms of the inverse (exponents): http://en.wikipedia.org/wiki/List_of...ler_operations

5. Here is a website with the rules for logarithms it may be helpfull

RULES OF LOGARITHMS