The function `f(x) = ( log ( pi +x))/( log ( e +x))` is a decreasing function in the interval `]0,oo[`. Is this statement true or false?

Answer

To determine whether the function \( f(x) = \frac{\log(\pi + x)}{\log(e + x)} \) is a decreasing function in the interval \( (0, \infty) \), we will follow these steps:

### Step 1: Differentiate the function
We need to find the derivative of \( f(x) \) to check if it is negative in the interval \( (0, \infty) \). We can use the quotient rule for differentiation.

The quotient rule states that if \( f(x) = \frac{u}{v} \), then:
\[ f'(x) = \frac{u'v - uv'}{v^2} \]
where \( u = \log(\pi + x) \) and \( v = \log(e + x) \).

### Step 2: Compute \( u' \) and \( v' \)
First, we differentiate \( u \) and \( v \):
- \( u = \log(\pi + x) \) implies \( u' = \frac{1}{\pi + x} \)
- \( v = \log(e + x) \) implies \( v' = \frac{1}{e + x} \)

### Step 3: Apply the quotient rule
Now we can apply the quotient rule:
\[ f'(x) = \frac{\left(\frac{1}{\pi + x}\right) \log(e + x) - \log(\pi + x) \left(\frac{1}{e + x}\right)}{(\log(e + x))^2} \]

### Step 4: Simplify the derivative
We can simplify the numerator:
\[ f'(x) = \frac{\log(e + x)}{(\pi + x)(\log(e + x))^2} - \frac{\log(\pi + x)}{(e + x)(\log(e + x))^2} \]
Combining the terms gives:
\[ f'(x) = \frac{(e + x) \log(e + x) - (\pi + x) \log(\pi + x)}{(e + x)(\pi + x)(\log(e + x))^2} \]

### Step 5: Analyze the sign of \( f'(x) \)
To determine if \( f(x) \) is decreasing, we need to check if \( f'(x) < 0 \). This means we need to analyze the numerator:
\[ (e + x) \log(e + x) < (\pi + x) \log(\pi + x) \]
for \( x > 0 \).

### Step 6: Check the inequality
We can check the inequality by substituting values in the interval \( (0, \infty) \):
- For small values of \( x \) (e.g., \( x = 1 \)):
- Left side: \( (e + 1) \log(e + 1) \)
- Right side: \( (\pi + 1) \log(\pi + 1) \)

As \( x \) increases, the left side grows slower than the right side due to the properties of logarithms and the constants involved.

### Conclusion
Since \( f'(x) < 0 \) for all \( x \in (0, \infty) \), we conclude that \( f(x) \) is a decreasing function in the interval \( (0, \infty) \).

Thus, the statement that the function \( f(x) \) is a decreasing function in the interval \( (0, \infty) \) is **true**.

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