# Thread: Comparing Two Logarithms Base 10

1. ## Comparing Two Logarithms Base 10

Hi all ---

For this multiple choice question, I don't know how to compare (a) and (b).

I try --- $\displaystyle \log \pi ? \sqrt{\log (\pi)^2}$

Then --- $\displaystyle \log \pi ? \sqrt{2\log\pi}$

or if I sub in L --- $\displaystyle L ? \sqrt{2L}$

Squaring both sides --- $\displaystyle L^2 ? 2L$

Because $\displaystyle 0 < L < 1$ --- squaring this for the 1st and then multiplying by 2 for the 2nd --- $\displaystyle 0 < L^2 < 1$ and $\displaystyle 0 < 2L < 2$.

1st Question --- But how can I compare the last two inequalities? I don't know what specifically L is?

2nd Question --- How does solution get $\displaystyle \sqrt{2L} > \sqrt{L \times L}$?

Thanks a lot ---

2. ## Re: Comparing Two Logarithms Base 10

$\displaystyle L = \log_{10}{\pi} < 1$

simplified choices in terms of L ...

(a) $\displaystyle L$ (b) $\displaystyle \sqrt{2L}$ (c) $\displaystyle \frac{1}{L^3}$ (d) $\displaystyle \frac{2}{L}$

for (a) and (b) ...

$\displaystyle L = \sqrt{L \cdot L} < \sqrt{2 \cdot L}$ because $\displaystyle 2 > L$

finally, since $\displaystyle L < 1$ , both (c) and (d) are > 1 > L ... you don't need to compare them against each other.