Certainly! Let's solve the equation step-by-step.
### Given Equation:
[tex]\[ 2 \cdot 3^{2x} = 9 \][/tex]
### Step 1: Simplify the right-hand side of the equation
We know that [tex]\(9\)[/tex] can be written as a power of [tex]\(3\)[/tex]:
[tex]\[ 9 = 3^2 \][/tex]
So the equation becomes:
[tex]\[ 2 \cdot 3^{2x} = 3^2 \][/tex]
### Step 2: Divide both sides by 2
Next, we'll isolate the exponential term by dividing both sides of the equation by [tex]\(2\)[/tex]:
[tex]\[ 3^{2x} = \frac{3^2}{2} \][/tex]
[tex]\[ 3^{2x} = \frac{9}{2} \][/tex]
[tex]\[ 3^{2x} = 4.5 \][/tex]
### Step 3: Take the natural logarithm of both sides to solve for [tex]\(x\)[/tex]
Taking the natural logarithm of both sides, we get:
[tex]\[ \ln(3^{2x}) = \ln(4.5) \][/tex]
Using the properties of logarithms, specifically [tex]\( \ln(a^b) = b \ln(a) \)[/tex], we can simplify this to:
[tex]\[ 2x \cdot \ln(3) = \ln(4.5) \][/tex]
Now, solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{\ln(4.5)}{2 \cdot \ln(3)} \][/tex]
Given the values:
[tex]\[ \ln(4.5) \approx 1.5040773967762742 \][/tex]
[tex]\[ \ln(3) \approx 1.0986122886681098 \][/tex]
Substitute these values into the equation:
[tex]\[ x = \frac{1.5040773967762742}{2 \cdot 1.0986122886681098} \][/tex]
[tex]\[ x = \frac{1.5040773967762742}{2.1972245773362196} \][/tex]
[tex]\[ x \approx 0.6845351232142712 \][/tex]
### Final Answer:
[tex]\[ x \approx 0.6845351232142712 \][/tex]
So, the solution to the equation [tex]\( 2 \cdot 3^{2x} = 9 \)[/tex] is approximately [tex]\( x \approx 0.6845 \)[/tex].
