If `log_(sqrt(5)) x + log_(5^(1//3)) x + log_(5^(1//4)) x + ...` up to 7 terms = 35, then the value of x is equal to

Options

A. 5
B. `5^(2)`
C. `5^(3)`
D. `sqrt(5)`

Answer

To solve the problem, we need to evaluate the expression given and find the value of \( x \).

### Step-by-Step Solution:

1. **Understanding the Expression**:
The expression given is:
\[ \log_{\sqrt{5}} x + \log_{5^{1/3}} x + \log_{5^{1/4}} x + \ldots \text{ (up to 7 terms)} = 35 \]
We need to rewrite each logarithm in terms of a common base.

2. **Changing the Base**:
We can use the change of base formula:
\[ \log_{a^b} c = \frac{1}{b} \log_a c \]
Therefore:
- \(\log_{\sqrt{5}} x = \frac{1}{1/2} \log_5 x = 2 \log_5 x\)
- \(\log_{5^{1/3}} x = \frac{1}{1/3} \log_5 x = 3 \log_5 x\)
- \(\log_{5^{1/4}} x = \frac{1}{1/4} \log_5 x = 4 \log_5 x\)
- Continuing this way, we can write the terms as:
- \(\log_{5^{1/5}} x = 5 \log_5 x\)
- \(\log_{5^{1/6}} x = 6 \log_5 x\)
- \(\log_{5^{1/7}} x = 7 \log_5 x\)

3. **Summing the Terms**:
Now we can sum these terms:
\[ 2 \log_5 x + 3 \log_5 x + 4 \log_5 x + 5 \log_5 x + 6 \log_5 x + 7 \log_5 x + 8 \log_5 x \]
This simplifies to:
\[ (2 + 3 + 4 + 5 + 6 + 7 + 8) \log_5 x \]

4. **Calculating the Sum**:
The sum of the integers from 2 to 8 is:
\[ 2 + 3 + 4 + 5 + 6 + 7 + 8 = 35 \]
Therefore, we have:
\[ 35 \log_5 x \]

5. **Setting Up the Equation**:
We set this equal to 35:
\[ 35 \log_5 x = 35 \]

6. **Dividing Both Sides**:
Dividing both sides by 35 gives:
\[ \log_5 x = 1 \]

7. **Finding x**:
To find \( x \), we rewrite the logarithmic equation in exponential form:
\[ x = 5^1 = 5 \]

### Final Answer:
Thus, the value of \( x \) is:
\[ \boxed{5} \]