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 \(3^{x+2} = 45\) in terms of \(a\) and \(b\), where \( \log_{10} 2 = a \) and \( \log_{10} 3 = b\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 3^{x+2} = 45 \]
We can express \(45\) as \(9 \times 5\) or \(3^2 \times 5\). Thus, we can rewrite the equation as:
\[ 3^{x+2} = 3^2 \times 5 \]

### Step 2: Isolate \(3^x\)
Dividing both sides by \(3^2\) gives us:
\[ 3^x = \frac{45}{9} = 5 \]

### Step 3: Take logarithm on both sides
Now, we take the logarithm (base 10) of both sides:
\[ \log_{10}(3^x) = \log_{10}(5) \]
Using the logarithmic property \(\log(a^b) = b \log(a)\), we can rewrite the left side:
\[ x \log_{10}(3) = \log_{10}(5) \]

### Step 4: Solve for \(x\)
Now, we can solve for \(x\):
\[ x = \frac{\log_{10}(5)}{\log_{10}(3)} \]
Since we know that \(\log_{10}(3) = b\), we can substitute:
\[ x = \frac{\log_{10}(5)}{b} \]

### Step 5: Express \(\log_{10}(5)\) in terms of \(a\) and \(b\)
Next, we need to express \(\log_{10}(5)\) in terms of \(a\) and \(b\). We can use the fact that \(5 = \frac{10}{2}\):
\[ \log_{10}(5) = \log_{10}\left(\frac{10}{2}\right) = \log_{10}(10) - \log_{10}(2) \]
Using the properties of logarithms, we know that \(\log_{10}(10) = 1\) and \(\log_{10}(2) = a\):
\[ \log_{10}(5) = 1 - a \]

### Step 6: Substitute back into the equation for \(x\)
Now we can substitute \(\log_{10}(5)\) back into our expression for \(x\):
\[ x = \frac{1 - a}{b} \]

### Final Result
Thus, the value of \(x\) in terms of \(a\) and \(b\) is:
\[ \boxed{\frac{1 - a}{b}} \]