Comparing Two Logarithms Base 10

**mathminor827** · September 11th 2011, 04:51 PM

Hi all ---

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

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

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

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

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

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

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 $\sqrt{2L} > \sqrt{L \times L}$ ?

Thanks a lot ---

**skeeter** · September 11th 2011, 05:59 PM

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

simplified choices in terms of L ...

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

for (a) and (b) ...

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

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

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