Question

Calculate the value of x from \(\frac{{log\left( {324} \right)}}{{log\left( {18} \right)}} = log\left( x \right)\).
1. 10
2. 100
3. 0
4. 36
5. None of these

Answer 1

Correct Answer - Option 2 : 100

Concept:

Formula of Logarithm:

\({a^b} = x\; \Leftrightarrow lo{g_a}x = b\), here a ≠ 1 and a > 0 and x be any number.

Properties of Logarithms:

1. \({\log _a}a = 1\)
2. \({\log _a}\left( {x.y} \right) = {\log _a}x + {\log _a}y\)
3. \({\log _a}\left( {\frac{x}{y}} \right) = {\log _a}x - {\log _a}y\)
4. \({\log _a}\left( {\frac{1}{x}} \right) = - {\log _a}x\)
5. \({\rm{lo}}{{\rm{g}}_a}{x^p} = p{\rm{lo}}{{\rm{g}}_a}x\)
6. \(lo{g_a}\left( x \right) = \frac{{lo{g_b}\left( x \right)}}{{lo{g_b}\left( a \right)}}\)

 

Calculation:

Given: \(\frac{{log\left( {324} \right)}}{{log\left( {18} \right)}} = log\left( x \right)\)

As we know that, \(\frac{{log\left( {324} \right)}}{{log\left( {18} \right)}} = \;\frac{{lo{g_{10}}\left( {324} \right)}}{{lo{g_{10}}\left( {18} \right)}}\) and \(log\left( x \right) = \;lo{g_{10}}\left( x \right)\)

Now, \(\frac{{lo{g_{10}}\left( {324} \right)}}{{lo{g_{10}}\left( {18} \right)}} = lo{g_{10}}\left( x \right)\)

Using the rule, \({a^b} = x\; \Leftrightarrow lo{g_a}x = b\) we have,

\({10^{\frac{{lo{g_{10}}\left( {324} \right)}}{{lo{g_{10}}\left( {18} \right)}}}} = x\)

\({10^{\frac{{lo{g_{10}}\left( {{{18}^2}} \right)}}{{lo{g_{10}}\left( {18} \right)}}}} = x\)

By the power rule we have,

\({10^{\frac{{2lo{g_{10}}\left( {18} \right)}}{{lo{g_{10}}\left( {18} \right)}}}} = x\)

⇒ x = 10²

⇒ x = 100

Hence, the value of x is 100.