To solve for [tex]\( x \)[/tex] in the equation
[tex]\[ 2^{x+2} = 9^{-3x}, \][/tex]
we will apply logarithms to both sides of the equation to make use of the properties of logarithms and solve for [tex]\( x \)[/tex].
1. First, rewrite the equation:
[tex]\[ 2^{x+2} = 9^{-3x} \][/tex]
2. Express 9 in terms of its base 3:
[tex]\[ 9^{-3x} = (3^2)^{-3x} = 3^{-6x} \][/tex]
3. Now the equation becomes:
[tex]\[ 2^{x+2} = 3^{-6x} \][/tex]
4. Take the logarithm of both sides. We will use logarithm base-10 ([tex]\( \log \)[/tex]):
[tex]\[ \log(2^{x+2}) = \log(3^{-6x}) \][/tex]
5. Apply the power rule of logarithms ([tex]\( \log(a^b) = b \log(a) \)[/tex]):
[tex]\[ (x + 2) \log(2) = (-6x) \log(3) \][/tex]
6. Distribute and collect terms involving [tex]\( x \)[/tex]:
[tex]\[ x \log(2) + 2 \log(2) = -6x \log(3) \][/tex]
7. Isolate [tex]\( x \)[/tex]:
[tex]\[ x \log(2) + 6x \log(3) = -2 \log(2) \][/tex]
[tex]\[ x (\log(2) + 6 \log(3)) = -2 \log(2) \][/tex]
8. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-2 \log(2)}{\log(2) + 6 \log(3)} \][/tex]
So, the exact answer is:
[tex]\[ x = \frac{-2 \log(2)}{\log(2) + 6 \log(3)} \][/tex]
In numerical terms, these logarithms simplify to:
[tex]\( \log(2) \approx 0.30103 \)[/tex] and [tex]\( \log(3) \approx 0.47712 \)[/tex].
Therefore, substituting these values in, we get:
[tex]\[ x \approx \frac{-2 \times 0.30103}{0.30103 + 6 \times 0.47712} \approx -0.1903 \][/tex]
The exact mathematical expression remains as:
[tex]\[ x = \frac{-2 \log(2)}{\log(2) + 6 \log(3)} \][/tex]
