Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Join our Q&A platform to get precise answers from experts in diverse fields and enhance your understanding. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.

Consider the equation [tex]\(6 \cdot e^{0.25 t}=9\)[/tex].

1. Solve the equation for [tex]\(t\)[/tex]. Express the solution as a logarithm in base-e.
[tex]\[
t = \square
\][/tex]

2. Approximate the value of [tex]\(t\)[/tex]. Round your answer to the nearest thousandth.
[tex]\[
t \approx \square
\][/tex]


Sagot :

To solve the equation [tex]\(6 \cdot e^{0.25 t} = 9\)[/tex] for [tex]\(t\)[/tex], follow these steps:

1. Isolate the exponential expression:
[tex]\[ 6 \cdot e^{0.25 t} = 9 \][/tex]
Divide both sides of the equation by 6:
[tex]\[ e^{0.25 t} = \frac{9}{6} \][/tex]
Simplify the fraction:
[tex]\[ e^{0.25 t} = 1.5 \][/tex]

2. Take the natural logarithm of both sides:
By applying the natural logarithm (denoted as [tex]\(\ln\)[/tex]) to both sides, we have:
[tex]\[ \ln(e^{0.25 t}) = \ln(1.5) \][/tex]

3. Utilize the properties of logarithms:
Recall that [tex]\(\ln(e^x) = x\)[/tex]. Therefore:
[tex]\[ 0.25 t = \ln(1.5) \][/tex]

4. Solve for [tex]\(t\)[/tex]:
To isolate [tex]\(t\)[/tex], divide both sides by 0.25:
[tex]\[ t = \frac{\ln(1.5)}{0.25} \][/tex]

Hence, the exact expression for [tex]\(t\)[/tex] is:
[tex]\[ t = \frac{\ln(1.5)}{0.25} \][/tex]

5. Approximate the value of [tex]\(t\)[/tex]:
Using a calculator to evaluate [tex]\(\frac{\ln(1.5)}{0.25}\)[/tex]:
[tex]\[ t \approx 1.622 \][/tex]

So, your final answers are:
[tex]\[ t = \frac{\ln(1.5)}{0.25} \][/tex]
[tex]\[ t \approx 1.622 \][/tex]
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.