If `log_(10)2,log_(10)(2^(x)-1),log_(10)(2^(x)+3)` are three consecutive terms of an AP, then which one of the following is correct?

Options

A. `x=0`
B. `x=1`
C. `x=log_(2)5`
D. `x=log_(5)2`

Explanation

To solve the problem, we need to find the value of \( x \) such that \( \log_{10} 2 \), \( \log_{10} (2^x - 1) \), and \( \log_{10} (2^x + 3) \) are three consecutive terms of an arithmetic progression (AP).

### Step 1: Set up the condition for AP
For three numbers \( A, B, C \) to be in AP, the condition is:
\[ 2B = A + C \]
In our case, let:
- \( A = \log_{10} 2 \)
- \( B = \log_{10} (2^x - 1) \)
- \( C = \log_{10} (2^x + 3) \)

Thus, we have:
\[ 2 \log_{10} (2^x - 1) = \log_{10} 2 + \log_{10} (2^x + 3) \]

### Step 2: Use properties of logarithms
Using the property of logarithms that states \( \log_a b + \log_a c = \log_a (bc) \), we can rewrite the equation:
\[ 2 \log_{10} (2^x - 1) = \log_{10} [2(2^x + 3)] \]

### Step 3: Simplify the equation
Now, we can express the left side using the power rule of logarithms:
\[ \log_{10} [(2^x - 1)^2] = \log_{10} [2(2^x + 3)] \]

Since the logarithms are equal, we can equate the arguments:
\[ (2^x - 1)^2 = 2(2^x + 3) \]

### Step 4: Expand and rearrange the equation
Expanding both sides:
\[ (2^x - 1)^2 = 2^{2x} - 2 \cdot 2^x + 1 \]
\[ 2(2^x + 3) = 2^{x+1} + 6 \]

Setting the two expressions equal gives:
\[ 2^{2x} - 2 \cdot 2^x + 1 = 2^{x+1} + 6 \]

### Step 5: Rearranging to form a quadratic equation
Rearranging the equation:
\[ 2^{2x} - 2^{x+1} - 2 \cdot 2^x + 1 - 6 = 0 \]
\[ 2^{2x} - 2^{x+1} - 2 \cdot 2^x - 5 = 0 \]

Let \( A = 2^x \). Then the equation becomes:
\[ A^2 - 2A - 2A - 5 = 0 \]
\[ A^2 - 4A - 5 = 0 \]

### Step 6: Factor the quadratic equation
Factoring the quadratic:
\[ (A - 5)(A + 1) = 0 \]

### Step 7: Solve for \( A \)
Setting each factor to zero gives:
\[ A - 5 = 0 \quad \Rightarrow \quad A = 5 \]
\[ A + 1 = 0 \quad \Rightarrow \quad A = -1 \quad \text{(not valid since } A = 2^x \text{ must be positive)} \]

Thus, we have:
\[ 2^x = 5 \]

### Step 8: Solve for \( x \)
Taking logarithm base 2 of both sides:
\[ x = \log_2 5 \]

### Conclusion
The value of \( x \) is:
\[ x = \log_2 5 \]