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Answer :
We start with the equation
[tex]$$1.69\, e^t = 31.5.$$[/tex]
Step 1. Isolate [tex]\(e^t\)[/tex]:
Divide both sides by [tex]\(1.69\)[/tex]:
[tex]$$
e^t = \frac{31.5}{1.69} \approx 18.6391.
$$[/tex]
Step 2. Take the natural logarithm:
Apply the natural logarithm on both sides to solve for [tex]\(t\)[/tex]:
[tex]$$
\ln(e^t) = \ln\left(\frac{31.5}{1.69}\right).
$$[/tex]
Since [tex]\(\ln(e^t) = t\)[/tex], we have:
[tex]$$
t = \ln\left(\frac{31.5}{1.69}\right) \approx 2.9253.
$$[/tex]
Thus, the solution rounded to four decimal places is
[tex]$$
t \approx 2.9253.
$$[/tex]
[tex]$$1.69\, e^t = 31.5.$$[/tex]
Step 1. Isolate [tex]\(e^t\)[/tex]:
Divide both sides by [tex]\(1.69\)[/tex]:
[tex]$$
e^t = \frac{31.5}{1.69} \approx 18.6391.
$$[/tex]
Step 2. Take the natural logarithm:
Apply the natural logarithm on both sides to solve for [tex]\(t\)[/tex]:
[tex]$$
\ln(e^t) = \ln\left(\frac{31.5}{1.69}\right).
$$[/tex]
Since [tex]\(\ln(e^t) = t\)[/tex], we have:
[tex]$$
t = \ln\left(\frac{31.5}{1.69}\right) \approx 2.9253.
$$[/tex]
Thus, the solution rounded to four decimal places is
[tex]$$
t \approx 2.9253.
$$[/tex]
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