Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Get immediate and reliable solutions to your questions from a community of experienced experts on our Q&A platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
Sure, let’s solve the equation [tex]\( 3^{x+2} = 15 \)[/tex] for [tex]\( x \)[/tex] using the change of base formula for logarithms.
1. Understand the equation:
We have [tex]\( 3^{x+2} = 15 \)[/tex]. Our goal is to isolate [tex]\( x \)[/tex].
2. Take the logarithm of both sides:
To simplify, we can take the natural logarithm (or any logarithm base) of both sides of the equation:
[tex]\[ \log(3^{x+2}) = \log(15) \][/tex]
3. Apply the power rule of logarithms:
The logarithm power rule states that [tex]\( \log(a^b) = b \log(a) \)[/tex]. Applying this to the left side:
[tex]\[ (x+2) \log(3) = \log(15) \][/tex]
4. Solve for [tex]\( x + 2 \)[/tex]:
To isolate [tex]\( x + 2 \)[/tex], divide both sides by [tex]\( \log(3) \)[/tex]:
[tex]\[ x+2 = \frac{\log(15)}{\log(3)} \][/tex]
5. Calculate the value:
Using the change of base formula [tex]\( \log_b(y) = \frac{\log(y)}{\log(b)} \)[/tex]:
[tex]\[ x+2 = \frac{\log(15)}{\log(3)} \][/tex]
The calculation of [tex]\( \frac{\log(15)}{\log(3)} \)[/tex] results in approximately [tex]\( 2.464973520717927 \)[/tex].
6. Isolate [tex]\( x \)[/tex]:
Now, subtract 2 from both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{\log(15)}{\log(3)} - 2 \][/tex]
Which yields:
[tex]\[ x \approx 2.464973520717927 - 2 \approx 0.4649735207179271 \][/tex]
So, the solution to the equation [tex]\( 3^{x+2} = 15 \)[/tex] for [tex]\( x \)[/tex] is approximately [tex]\( 0.465 \)[/tex].
Therefore, the closest answer choice is:
[tex]\[ \boxed{0.465} \][/tex]
1. Understand the equation:
We have [tex]\( 3^{x+2} = 15 \)[/tex]. Our goal is to isolate [tex]\( x \)[/tex].
2. Take the logarithm of both sides:
To simplify, we can take the natural logarithm (or any logarithm base) of both sides of the equation:
[tex]\[ \log(3^{x+2}) = \log(15) \][/tex]
3. Apply the power rule of logarithms:
The logarithm power rule states that [tex]\( \log(a^b) = b \log(a) \)[/tex]. Applying this to the left side:
[tex]\[ (x+2) \log(3) = \log(15) \][/tex]
4. Solve for [tex]\( x + 2 \)[/tex]:
To isolate [tex]\( x + 2 \)[/tex], divide both sides by [tex]\( \log(3) \)[/tex]:
[tex]\[ x+2 = \frac{\log(15)}{\log(3)} \][/tex]
5. Calculate the value:
Using the change of base formula [tex]\( \log_b(y) = \frac{\log(y)}{\log(b)} \)[/tex]:
[tex]\[ x+2 = \frac{\log(15)}{\log(3)} \][/tex]
The calculation of [tex]\( \frac{\log(15)}{\log(3)} \)[/tex] results in approximately [tex]\( 2.464973520717927 \)[/tex].
6. Isolate [tex]\( x \)[/tex]:
Now, subtract 2 from both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{\log(15)}{\log(3)} - 2 \][/tex]
Which yields:
[tex]\[ x \approx 2.464973520717927 - 2 \approx 0.4649735207179271 \][/tex]
So, the solution to the equation [tex]\( 3^{x+2} = 15 \)[/tex] for [tex]\( x \)[/tex] is approximately [tex]\( 0.465 \)[/tex].
Therefore, the closest answer choice is:
[tex]\[ \boxed{0.465} \][/tex]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.