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Aptitude::Logarithm

Home > Quantitative Aptitude > Logarithm > Subjective Solved Examples

Example 1 / 1

Evaluate
(i) \(\log_{2}128\) = ?
(ii) \(\log_{3}27\) = ?
(iii) \(\log_{7}\left(\frac{1}{343}\right)\) = ?
(iv) \(log_{100}\left(0.0001\right)\) = ?
(i) Let \(\log_{2}128\) = n
so, \(2^{n}\) = 128
Formula used: \(a^{m}\) = x \(\Leftrightarrow\) m = \(\log_{a}x\)
=> \(2^{n}\) = \(2^{7}\) or n = 7
=> \(\log_{2}128\) = 7 Ans.

(ii) Let \(\log_{3}27\) = n
so, \(3^{n}\) = 27
Formula used: \(a^{m}\) = x \(\Leftrightarrow\) m = \(\log_{a}x\)
=> \(3^{n}\) = \(3^{3}\) or n = 3
=> \(\log_{3}27\) = 3 Ans.

(iii) Let \(\log_{7}\left(\frac{1}{343}\right)\) = n
so, \(7^{n}\) = \(\frac{1}{343}\)
Formula used: \(a^{m}\) = x \(\Leftrightarrow\) m = \(\log_{a}x\)
=> \(7^{n}\) = \(\frac{1}{7^{3}}\)
=> \(7^{n}\) = \(7^{-3}\)or n = -3
=> \(\log_{7}\left(\frac{1}{343}\right)\) = -3 Ans.

(iv) Let \(log_{100}\left(0.0001\right)\) = n
so, \(100^{n}\) = 0.0001
Formula used: \(a^{m}\) = x \(\Leftrightarrow\) m = \(\log_{a}x\)
=> \(100^{n}\) = \(\frac{1}{10000}\)
=> \(100^{n}\) = \(\frac{1}{100^{2}}\)
=> \(100^{n}\) = \(100^{-2}\) or n = -2
\(log_{100}\left(0.001\right)\) = -2 Ans.
NA
NA
(i) Simplify like below:
\(\log_{2}128\) = \(\log_{2}2^{7}\) = \(7 \log_{2}2^{7}\) = 7
Formula used:
\(\log_{x}x\) = 1 AND \(\log_{x}x^m\) = \(m \log_{x}x^m\)

(ii) Simplify like below:
\(\log_{3}27\) = \(\log_{3}3^{3}\) = 3
Formula used:
\(\log_{x}x\) = 1 AND \(\log_{x}x^m\) = \(m \log_{x}x^m\)

(iii) Simplify like below:
\(\log_{7}\left(\frac{1}{343}\right)\) = \(\log_{7}\left(\frac{1}{7^{3}}\right)\)
= \(\log_{7}\left(7^{-3}\right)\) = -3
Formula used:
\(\log_{x}x\) = 1 AND \(\log_{x}x^m\) = \(m \log_{x}x^m\)

(iv) Simplify like below:
Let \(log_{100}\left(0.0001\right)\) = \(log_{100}\left(\frac{1}{10000}\right)\)
= \(log_{100}\left(\frac{1}{100^{2}}\right)\) = \(log_{100}\left(100^{-2}\right)\) = -2
Formula used:
\(\log_{x}x\) = 1 AND \(\log_{x}x^m\) = \(m \log_{x}x^m\)
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