To solve the equation [tex]\(3^{2x+3} = 243\)[/tex] for [tex]\(x\)[/tex], let's follow a step-by-step approach.
1. Express 243 as a power of 3: To make the equation easier to handle, write 243 as a power of 3. We know that [tex]\(243 = 3^5\)[/tex], so we can rewrite the equation as:
[tex]\[
3^{2x + 3} = 3^5
\][/tex]
2. Set the exponents equal to each other: Since the bases are the same (both are 3), we can set the exponents equal to each other:
[tex]\[
2x + 3 = 5
\][/tex]
3. Solve for [tex]\(x\)[/tex]: Now, we solve the equation [tex]\(2x + 3 = 5\)[/tex] for [tex]\(x\)[/tex].
- Subtract 3 from both sides:
[tex]\[
2x + 3 - 3 = 5 - 3
\][/tex]
Simplifying this gives:
[tex]\[
2x = 2
\][/tex]
- Divide both sides by 2:
[tex]\[
\frac{2x}{2} = \frac{2}{2}
\][/tex]
Simplifying this gives:
[tex]\[
x = 1
\][/tex]
So, the value of [tex]\(x\)[/tex] that satisfies the equation [tex]\(3^{2x+3} = 243\)[/tex] is [tex]\(x = 1\)[/tex].
The correct answer is:
D. [tex]\(x = 1\)[/tex]
