If `"log"_(8)x = 25 " and log"_(2) y = 50`, then x =

Options

A. `y^(3//2)`
B. 2y
C. y
D. `(y)/(2)`

Answer

To solve the given problem, we will follow these steps:

### Step 1: Rewrite the logarithmic equation
We start with the equation given in the problem:
\[ \log_{8} x = 25 \]
This can be rewritten in exponential form:
\[ x = 8^{25} \]

### Step 2: Express 8 in terms of base 2
Next, we express 8 as a power of 2:
\[ 8 = 2^3 \]
Substituting this into our equation for \(x\):
\[ x = (2^3)^{25} \]

### Step 3: Simplify the exponent
Using the power of a power property \((a^m)^n = a^{m \cdot n}\), we simplify:
\[ x = 2^{3 \cdot 25} = 2^{75} \]

### Step 4: Use the second logarithmic equation
Now, we look at the second equation given in the problem:
\[ \log_{2} y = 50 \]
This can also be rewritten in exponential form:
\[ y = 2^{50} \]

### Step 5: Relate \(x\) to \(y\)
We have \(x = 2^{75}\) and \(y = 2^{50}\). To express \(x\) in terms of \(y\), we can write:
\[ x = 2^{75} = 2^{50 \cdot \frac{75}{50}} = (2^{50})^{\frac{75}{50}} = y^{\frac{75}{50}} \]
Simplifying \(\frac{75}{50}\):
\[ \frac{75}{50} = \frac{3}{2} \]
Thus, we have:
\[ x = y^{\frac{3}{2}} \]

### Final Answer
So, the value of \(x\) in terms of \(y\) is:
\[ \boxed{y^{\frac{3}{2}}} \]

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