Question

If \(\frac{{\log x}}{{\log 3}} = \frac{{\log 64}}{{\log 4}} = \frac{{\log 64}}{{\log y}}\) find the value of x and y.
1. 27, 27
2. 4, 27
3. 27, 4
4. 25, 5
5. None of these

Answer 1

Correct Answer - Option 3 : 27, 4

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: \(\frac{{\log x}}{{\log 3}} = \frac{{\log 64}}{{\log 4}} = \frac{{\log 64}}{{\log y}}\)

Let us first consider, \(\frac{{\log x}}{{\log 3}} = \frac{{\log 64}}{{\log 4}}\)

Here, \(\log 64 = \log {4^3}\)

Using the power rule, \(\log 64 = 3\log 4\).

\(\Rightarrow \frac{{\log 64}}{{\log 4}} = \frac{{3\log 4}}{{\log 4}} = 3\)

Thus, \(\frac{{\log x}}{{\log 3}} = 3\)

By the base rule of logarithms, \(\frac{{\log x}}{{\log 3}} = {\log _3}x\)

Therefore, \({\log _3}x = 3\) by using \({a^b} = x\; \cdot lo{g_a}x = b\) rule we get

⟹ x = 3³ = 27

Hence, x = 27

Now consider, \(\frac{{\log 64}}{{\log 4}} = \frac{{\log 64}}{{\log y}}\)

\(\Rightarrow \frac{1}{{\log 4}} = \frac{1}{{\log y}}\)

\(\Rightarrow \log 4 = \log y\)

Using \({a^b} = x \Leftrightarrow lo{g_a}x = b\), we get

\(\Rightarrow {10^{\log 4}} = {10^{\log y}}\)

⟹ y = 4

Hence, the value of x and y is 27 and 4.