If `a!= 1` and `In a^2 + (In a^2)^2 + (In a^2)^3 + ... = 3 (ln a + (lna)^2 + (lna)^3 + (lna)^4+...)` then 'a' is equal to

Answer

To solve the equation given in the problem, we start by rewriting the infinite series on both sides of the equation.

### Step 1: Write the equation
We have:
\[ \ln a^2 + (\ln a^2)^2 + (\ln a^2)^3 + \ldots = 3 \left( \ln a + (\ln a)^2 + (\ln a)^3 + (\ln a)^4 + \ldots \right) \]

### Step 2: Use properties of logarithms
Using the property of logarithms, we can rewrite \(\ln a^2\) as:
\[ \ln a^2 = 2 \ln a \]
Thus, the left-hand side becomes:
\[ 2 \ln a + (2 \ln a)^2 + (2 \ln a)^3 + \ldots \]

### Step 3: Identify the series as a geometric series
The left-hand side is a geometric series with first term \(2 \ln a\) and common ratio \(2 \ln a\). The sum of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
where \(a\) is the first term and \(r\) is the common ratio. Therefore, we have:
\[ \text{Left-hand side} = \frac{2 \ln a}{1 - 2 \ln a} \]

### Step 4: Analyze the right-hand side
The right-hand side is also a geometric series with first term \(\ln a\) and common ratio \(\ln a\):
\[ \text{Right-hand side} = 3 \left( \frac{\ln a}{1 - \ln a} \right) \]

### Step 5: Set the two sides equal
Now we set the left-hand side equal to the right-hand side:
\[ \frac{2 \ln a}{1 - 2 \ln a} = 3 \left( \frac{\ln a}{1 - \ln a} \right) \]

### Step 6: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ 2 \ln a (1 - \ln a) = 3 \ln a (1 - 2 \ln a) \]

### Step 7: Expand both sides
Expanding both sides results in:
\[ 2 \ln a - 2 (\ln a)^2 = 3 \ln a - 6 (\ln a)^2 \]

### Step 8: Rearrange the equation
Rearranging gives:
\[ 4 (\ln a)^2 - \ln a = 0 \]

### Step 9: Factor the equation
Factoring out \(\ln a\):
\[ \ln a (4 \ln a - 1) = 0 \]

### Step 10: Solve for \(\ln a\)
This gives us two solutions:
1. \(\ln a = 0\) (which implies \(a = 1\), but we are given \(a \neq 1\))
2. \(4 \ln a - 1 = 0\) which leads to:
\[ \ln a = \frac{1}{4} \]

### Step 11: Exponentiate to find \(a\)
Exponentiating both sides gives:
\[ a = e^{\frac{1}{4}} \]

### Final Answer
Thus, the value of \(a\) is:
\[ a = e^{\frac{1}{4}} \]

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