`4^(log_9 3)+9^(log_2 4)=1 0^(log_x 83)`, then x is equal to

Options

A. 2
B. 3
C. 10
D. 30

Explanation

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 logarithms
We can express \( 9 \) and \( 4 \) in terms of base \( 3 \) and \( 2 \) respectively:
- \( 9 = 3^2 \)
- \( 4 = 2^2 \)

Thus, we can rewrite the terms:
\[ 4^{\log_9 3} = 4^{\frac{\log_3 3}{\log_3 9}} = 4^{\frac{1}{2}} = 2 \]
and
\[ 9^{\log_2 4} = 9^{\frac{\log_2 4}{\log_2 9}} = 9^{\frac{2}{\log_2 3^2}} = 9^{\frac{2}{2 \log_2 3}} = 9^{\frac{1}{\log_2 3}} = 3^2 = 81 \]

### Step 2: Combine the results
Now we can combine the results from Step 1:
\[ 2 + 81 = 83 \]

### Step 3: Set the equation
Now we have:
\[ 83 = 10^{\log_x 83} \]

### Step 4: Use the property of logarithms
Using the property \( a^{\log_a b} = b \), we can rewrite this as:
\[ 83 = x^{\log_{10} 83} \]

### Step 5: Solve for \( x \)
To isolate \( x \), we can take logarithm base 10 on both sides:
\[ \log_{10} 83 = \log_{10} (x^{\log_{10} 83}) = \log_{10} 83 \cdot \log_{10} x \]
This implies:
\[ 1 = \log_{10} x \]
Thus, we can conclude:
\[ x = 10 \]

### Final Answer
The value of \( x \) is \( 10 \).

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