To solve the equation [tex]\(3^{x+2} = 15\)[/tex] for [tex]\(x\)[/tex], we can use logarithms and the change of base formula. Here's a step-by-step solution:
1. Start with the given equation:
[tex]\[ 3^{x+2} = 15 \][/tex]
2. Take the logarithm of both sides. You can choose any logarithm base, but in this case, we'll use the natural logarithm (ln) for simplicity. So we take the natural logarithm ([tex]\(\ln\)[/tex]) of both sides:
[tex]\[ \ln(3^{x+2}) = \ln(15) \][/tex]
3. Use the logarithm power rule, [tex]\(\ln(a^b) = b \ln(a)\)[/tex], to move the exponent in front of the logarithm:
[tex]\[ (x+2) \ln(3) = \ln(15) \][/tex]
4. Solve for [tex]\(x+2\)[/tex] by dividing both sides by [tex]\(\ln(3)\)[/tex]:
[tex]\[ x+2 = \frac{\ln(15)}{\ln(3)} \][/tex]
5. Now, calculate the values:
- [tex]\(\ln(15) \approx 2.708\)[/tex]
- [tex]\(\ln(3) \approx 1.099\)[/tex]
So,
[tex]\[ x+2 \approx \frac{2.708}{1.099} \approx 2.465 \][/tex]
6. Subtract 2 from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[ x \approx 2.465 - 2 \approx 0.465 \][/tex]
Thus, the solution for the equation [tex]\(3^{x+2} = 15\)[/tex] is [tex]\(x \approx 0.465\)[/tex].
Among the given choices,
- [tex]\( -1.594 \)[/tex]
- [tex]\( \mathbf{0.465} \)[/tex] (this is the correct answer)
- [tex]\( 2.406 \)[/tex]
- [tex]\( 4.465 \)[/tex]
The correct answer is [tex]\(\boldsymbol{0.465}\)[/tex].
