Which of the correct order for a given number `alpha` in increasing order

Options

A. `log_(2)alpha`, `log_(3)alpha`, `log_(e )alpha`, `log_(10)alpha`
B. `log_(10)alpha`, `log_(3)alpha`, `log_(e )alpha`, `log_(2)alpha`
C. `log_(10)alpha`, `log_(2)alpha`, `log_(e )alpha`, `log_(3)alpha`
D. `log_(3)alpha`, `log_(e)alpha`, `log_(2)alpha`, `log_(1)alpha`

Answer

To solve the problem of determining the correct order of logarithmic values for a given number \( \alpha \) in increasing order, we can follow these steps:

### Step 1: Understand the properties of logarithms
The logarithm function \( \log_b(a) \) is defined as the exponent to which the base \( b \) must be raised to produce the number \( a \). When the base \( b \) increases, the value of \( \log_b(a) \) decreases for a fixed \( a \).

### Step 2: Identify the bases
We are given several bases: 2, 3, \( e \) (Euler's number, approximately 2.718), and 10. We need to find the order of \( \log_2(\alpha) \), \( \log_3(\alpha) \), \( \log_e(\alpha) \), and \( \log_{10}(\alpha) \).

### Step 3: Compare the logarithmic values
Since \( \alpha \) is fixed, we can compare the logarithmic values based on the properties of logarithms:

1. **Base 10**: \( \log_{10}(\alpha) \)
2. **Base \( e \)**: \( \log_e(\alpha) \)
3. **Base 3**: \( \log_3(\alpha) \)
4. **Base 2**: \( \log_2(\alpha) \)

### Step 4: Determine the order
Using the property that larger bases yield smaller logarithmic values:
- \( \log_{10}(\alpha) \) will be the smallest since 10 is the largest base.
- \( \log_e(\alpha) \) will be next since \( e \) is smaller than 10 but larger than 3.
- \( \log_3(\alpha) \) will follow since 3 is smaller than \( e \).
- \( \log_2(\alpha) \) will be the largest since 2 is the smallest base.

Thus, the increasing order of the logarithmic values is:
\[ \log_{10}(\alpha) < \log_e(\alpha) < \log_3(\alpha) < \log_2(\alpha) \]

### Step 5: Write the final answer
The correct order in increasing order is:
\[ \log_{10}(\alpha), \log_e(\alpha), \log_3(\alpha), \log_2(\alpha) \]