The value 'x' satisfying the equation, `4^(log_(9)3)+9^(log_(2)4)=10^(log_(x)83)is ____`

Answer

To solve the equation \( 4^{\log_{9}3} + 9^{\log_{2}4} = 10^{\log_{x}83} \), we will follow these steps:

### Step 1: Rewrite the bases
We can rewrite \( 4 \) and \( 9 \) in terms of powers of \( 2 \) and \( 3 \):
- \( 4 = 2^2 \)
- \( 9 = 3^2 \)

Thus, we can rewrite the equation as:
\[ (2^2)^{\log_{9}3} + (3^2)^{\log_{2}4} = 10^{\log_{x}83} \]

### Step 2: Apply the power rule of logarithms
Using the power rule of exponents, we can simplify the left-hand side:
\[ 2^{2 \cdot \log_{9}3} + 3^{2 \cdot \log_{2}4} = 10^{\log_{x}83} \]

### Step 3: Simplify the logarithmic expressions
Using the change of base formula for logarithms:
\[ \log_{9}3 = \frac{\log_{3}3}{\log_{3}9} = \frac{1}{2} \quad \text{(since } \log_{3}9 = 2\text{)} \]
\[ \log_{2}4 = \frac{\log_{2}4}{\log_{2}2} = 2 \quad \text{(since } \log_{2}2 = 1\text{)} \]

Substituting these values back into the equation:
\[ 2^{2 \cdot \frac{1}{2}} + 3^{2 \cdot 2} = 10^{\log_{x}83} \]
This simplifies to:
\[ 2^1 + 3^4 = 10^{\log_{x}83} \]

### Step 4: Calculate the left-hand side
Calculating the left-hand side:
\[ 2 + 81 = 10^{\log_{x}83} \]
Thus, we have:
\[ 83 = 10^{\log_{x}83} \]

### Step 5: Take logarithm on both sides
Taking logarithm base \( 10 \) on both sides:
\[ \log_{10}83 = \log_{10}(10^{\log_{x}83}) \]
Using the property of logarithms:
\[ \log_{10}83 = \log_{x}83 \cdot \log_{10}10 \]
Since \( \log_{10}10 = 1 \), we have:
\[ \log_{10}83 = \log_{x}83 \]

### Step 6: Solve for \( x \)
From the equality of logarithms, we can deduce that:
\[ x = 10 \]

Thus, the value of \( x \) satisfying the equation is:
\[ \boxed{10} \]