If `(logx)/(log5) = (log 36)/(log 6) = (log 64)/(log y)` what are the values of x and y respectively ?

Options

A. 8&25
B. 25&8
C. 8.8
D. 25.25

Answer

To solve the equation \(\frac{\log x}{\log 5} = \frac{\log 36}{\log 6} = \frac{\log 64}{\log y}\), we will find the values of \(x\) and \(y\) step by step.

### Step 1: Simplify \(\frac{\log 36}{\log 6}\)

We know that \(36 = 6^2\). Therefore, we can rewrite \(\log 36\) as:

\[ \log 36 = \log(6^2) = 2 \log 6 \]

Now substituting this back into the equation, we have:

\[ \frac{\log 36}{\log 6} = \frac{2 \log 6}{\log 6} = 2 \]

### Step 2: Set \(\frac{\log x}{\log 5}\) equal to 2

From the first part of the equation, we can write:

\[ \frac{\log x}{\log 5} = 2 \]

Multiplying both sides by \(\log 5\):

\[ \log x = 2 \log 5 \]

### Step 3: Rewrite \(\log x\) using properties of logarithms

Using the property of logarithms that states \(\log a^b = b \log a\), we can rewrite \(2 \log 5\) as:

\[ \log x = \log(5^2) = \log 25 \]

### Step 4: Solve for \(x\)

Since the logarithms are equal, we can equate the arguments:

\[ x = 25 \]

### Step 5: Simplify \(\frac{\log 64}{\log y}\)

Next, we simplify \(\frac{\log 64}{\log y}\). We know that \(64 = 8^2\), so:

\[ \log 64 = \log(8^2) = 2 \log 8 \]

Now substituting this into the equation, we have:

\[ \frac{\log 64}{\log y} = \frac{2 \log 8}{\log y} \]

### Step 6: Set \(\frac{\log 64}{\log y}\) equal to 2

From the second part of the equation, we can write:

\[ \frac{2 \log 8}{\log y} = 2 \]

Multiplying both sides by \(\log y\):

\[ 2 \log 8 = 2 \log y \]

### Step 7: Divide both sides by 2

Dividing both sides by 2 gives us:

\[ \log 8 = \log y \]

### Step 8: Solve for \(y\)

Since the logarithms are equal, we can equate the arguments:

\[ y = 8 \]

### Final Values

Thus, the values of \(x\) and \(y\) are:

\[ x = 25, \quad y = 8 \]