Given `log_(10)2 = a` and `log_(10)3 = b`. If `3^(x+2) = 45`, then the value of x in terms of a and b is-

Options

A. `(a-1)/(b)`
B. `(1-a)/(b)`
C. `(1+a)/(b)`
D. `(b)/(1-a)`

Answer

To solve the equation \(3^{x+2} = 45\) and express \(x\) in terms of \(a\) and \(b\) where \( \log_{10} 2 = a\) and \( \log_{10} 3 = b\), follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ 3^{x+2} = 45 \]

### Step 2: Isolate \(3^x\)
We can rewrite \(45\) as \(9 \times 5\) or \(3^2 \times 5\):
\[ 3^{x+2} = 3^2 \times 5 \]
This implies:
\[ 3^{x+2} = 3^2 \cdot 5 \]
Now, we can express \(3^{x}\) as:
\[ 3^x = \frac{45}{3^2} = 5 \]

### Step 3: Take logarithm of both sides
Taking logarithm base 10 on both sides:
\[ \log_{10}(3^x) = \log_{10}(5) \]

### Step 4: Use logarithm properties
Using the property of logarithms \( \log_{10}(a^b) = b \cdot \log_{10}(a)\):
\[ x \cdot \log_{10}(3) = \log_{10}(5) \]

### Step 5: Solve for \(x\)
Now, isolate \(x\):
\[ x = \frac{\log_{10}(5)}{\log_{10}(3)} \]

### Step 6: Express \(\log_{10}(5)\) in terms of \(a\) and \(b\)
We can express \(\log_{10}(5)\) using the change of base formula:
\[ \log_{10}(5) = \log_{10}\left(\frac{10}{2}\right) = \log_{10}(10) - \log_{10}(2) = 1 - a \]
Thus:
\[ x = \frac{1 - a}{b} \]

### Final Answer
The value of \(x\) in terms of \(a\) and \(b\) is:
\[ x = \frac{1 - a}{b} \]