If $a = 1 + l o g_{10} 2 - l o g_{10} 5, b = 2 l o g_{10} 3$ and $c = l o g_{10} m - l o g_{10} 5,$ Find the value of m if a + b = 2c

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Solution

The correct option is A 30
We have,
$a = 1 + l o g_{10} 2 - l o g_{10} 5 = l o g_{10} 10 + l o g_{10} 2 - l o g_{10} 5 = l o g_{10} (\frac{10 \times 2}{5}) = l o g_{10} 4$
$b = 2 l o g_{10} 3 = l o g_{10} 9$
$c = l o g_{10} m - l o g_{10} 5 = l o g_{10} (\frac{m}{5})$
So, a + b = 2c
$\Rightarrow l o g_{10} 4 + l o g_{10} 9 = 2 l o g_{10} (\frac{m}{5})$
$\Rightarrow l o g_{10} (\frac{m^{2}}{25}) = l o g_{10} 36$
$\Rightarrow m^{2} = 36 \times 25$
$\Rightarrow m = \pm 30$
As argument of log cannot be negative.
So, option a is correct.

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