Question
Solve the equation algebraically and check graphically.
e Superscript tequals803
Question content area bottom
Part 1
tequals
  
enter your response here
​(Type an integer or decimal rounded to three decimal places as​ needed.)
e Superscript tequals803
Question content area bottom
Part 1
tequals
  
enter your response here
​(Type an integer or decimal rounded to three decimal places as​ needed.)
Answers
Answered by
GPT-4o mini
To solve the equation \( e^t = 803 \) algebraically, we can take the natural logarithm (ln) of both sides. The natural logarithm and the exponential function are inverses of each other.
Starting with the equation:
\[
e^t = 803
\]
Now, we take the natural logarithm of both sides:
\[
\ln(e^t) = \ln(803)
\]
Using the property of logarithms that states \( \ln(e^x) = x \), we get:
\[
t = \ln(803)
\]
Now, we can calculate \( \ln(803) \) using a calculator:
\[
t \approx \ln(803) \approx 6.684
\]
So the solution, rounded to three decimal places, is:
\[
t \approx 6.684
\]
### Graphical Check
To check graphically, we can plot the functions \( y = e^t \) and \( y = 803 \). The point where these two graphs intersect is the value of \( t \).
1. The graph of \( y = e^t \) is an exponential curve that increases rapidly.
2. The line \( y = 803 \) is a horizontal line.
You can use graphing software or tools (like Desmos, GeoGebra, or a graphing calculator) to visualize these functions and identify their intersection.
When you graph these functions, the intersection point will occur at approximately \( t \approx 6.684 \), affirming our algebraic solution.
Thus, the final answer is:
\[
t \approx 6.684
\]
Starting with the equation:
\[
e^t = 803
\]
Now, we take the natural logarithm of both sides:
\[
\ln(e^t) = \ln(803)
\]
Using the property of logarithms that states \( \ln(e^x) = x \), we get:
\[
t = \ln(803)
\]
Now, we can calculate \( \ln(803) \) using a calculator:
\[
t \approx \ln(803) \approx 6.684
\]
So the solution, rounded to three decimal places, is:
\[
t \approx 6.684
\]
### Graphical Check
To check graphically, we can plot the functions \( y = e^t \) and \( y = 803 \). The point where these two graphs intersect is the value of \( t \).
1. The graph of \( y = e^t \) is an exponential curve that increases rapidly.
2. The line \( y = 803 \) is a horizontal line.
You can use graphing software or tools (like Desmos, GeoGebra, or a graphing calculator) to visualize these functions and identify their intersection.
When you graph these functions, the intersection point will occur at approximately \( t \approx 6.684 \), affirming our algebraic solution.
Thus, the final answer is:
\[
t \approx 6.684
\]