# Thread: Which quantitiy is larger?

1. ## Which quantitiy is larger?

.

Show which of the following quantities is larger:

$\displaystyle log_2{3} \ \ or \ \ log_3{5}$

Demonstrate it. You may not use a calculator, computer, or logarithm tables.

You may not use calculus, in the sense that you may not use derivatives or antiderivatives.

You may use arithmetic and high school algebra to include the rules of exponents and logarithms.

2. ## Re: Which quantitiy is larger?

You may use one of following rules to solve this.
$\log_a b = \frac{1}{\log_b a}$ or $\log_a b = \frac{\log_c b}{\log_c a}$
The idea is that write both terms in a same base (here it is 3).
Can you figure out the rest?

3. ## Re: Which quantitiy is larger?

Originally Posted by zemozamster
Can you figure out the rest?
He's not asking a question. It's a challenge problem.

4. ## Re: Which quantitiy is larger?

Originally Posted by romsek
He's not asking a question. It's a challenge problem.
Did not notice the thread category.

5. ## Re: Which quantitiy is larger?

Originally Posted by zemozamster
You may use one of following rules to solve this.
$\log_a b = \frac{1}{\log_b a}$ or $\log_a b = \frac{\log_c b}{\log_c a}$

> > > The idea is that write both terms in a same base (here it is 3). < < <

Can you figure out the rest?
Put aside that you didn't realize I was giving a challenge problem, even though I posted under "Math Challenge Problems."

$\displaystyle \dfrac{log_3{3}}{log_3{2}} \ \ \ versus \ \ \ \dfrac{log_3{5}}{log_3{3}}$

$\displaystyle \dfrac{1}{log_3{2}} \ \ \ versus \ \ \ \dfrac{log_3{5}}{1}$ **

Did you intend the above at one point in your steps? Suppose it is so:

I could rewrite it as:

$\displaystyle 1 \ \ \ versus \ \ \ (log_3{2})(log_3{5})$

The right-hand side is a product of a positive quantity that is less than one with a positive quantity that is greater than one,
but I don't know their relative sizes from that to make a determination of the size of that product relative to 1. Where would
you go from there?

** If you did get to this step, what did you do next?

By the way, my different demonstration among many can be shown in this thread at a later time.

6. ## Re: Which quantitiy is larger?

Spoiler:
$9 > 8 \Rightarrow 2\log_2(3) > 3$

$27 > 25 \Rightarrow 3 > 2\log_3(5)$

so

$2\log_3(5) < 3 < 2\log_2(3)$

$\log_3(5) < \log_2(3)$

7. ## Re: Which quantitiy is larger?

Originally Posted by romsek
Spoiler:
$9 > 8 \Rightarrow 2\log_2(3) > 3$

$27 > 25 \Rightarrow 3 > 2\log_3(5)$

so

$2\log_3(5) < 3 < 2\log_2(3)$

$\log_3(5) < \log_2(3)$
The type of solution given by romsek that doesn't rely on arithmetic calculations by hand, such as possibly
multiplying out decimals, is not a requirement, but it is relatively elegant. I thank romsek for this solution.

8. ## Re: Which quantitiy is larger?

Originally Posted by greg1313
but it is relatively elegant.
your solution must be a single alien symbol

9. ## Re: Which quantitiy is larger?

Originally Posted by romsek
your solution must be a single alien symbol
My solution is a variation of yours, but yours is more concise, and the steps have more of a natural flow than do mine.
I prefer yours over mine.