To solve the equation [tex]\( e^{5t} = 27 \)[/tex] for [tex]\( t \)[/tex], follow these steps:
1. Understand the equation:
The equation is [tex]\( e^{5t} = 27 \)[/tex]. Here, [tex]\( e \)[/tex] is the base of the natural logarithm.
2. Take the natural logarithm on both sides:
Taking the natural logarithm ([tex]\(\ln\)[/tex]) on both sides helps to eliminate the exponential. Thus, we get:
[tex]\[
\ln(e^{5t}) = \ln(27)
\][/tex]
3. Simplify the left-hand side:
By using the property of logarithms [tex]\(\ln(e^x) = x\)[/tex], we can simplify the left-hand side:
[tex]\[
5t = \ln(27)
\][/tex]
4. Solve for [tex]\( t \)[/tex]:
To isolate [tex]\( t \)[/tex], divide both sides of the equation by 5:
[tex]\[
t = \frac{\ln(27)}{5}
\][/tex]
5. Express [tex]\( \ln(27) \)[/tex] using known properties of logarithms:
Recognize that [tex]\( 27 \)[/tex] can be written as [tex]\( 3^3 \)[/tex]. Thus:
[tex]\[
\ln(27) = \ln(3^3)
\][/tex]
Using the logarithm property [tex]\( \ln(a^b) = b \ln(a) \)[/tex], we get:
[tex]\[
\ln(3^3) = 3 \ln(3)
\][/tex]
Substitute this back into the equation for [tex]\( t \)[/tex]:
[tex]\[
t = \frac{3 \ln(3)}{5}
\][/tex]
6. Finish up the solution:
Therefore, the simplified expression for [tex]\( t \)[/tex] is:
[tex]\[
t = \frac{3}{5} \ln(3)
\][/tex]
However, it's important to remember that logarithms with complex numbers might yield additional solutions involving imaginary components. Taking the logarithms in the complex plane, we have to include possible imaginary components.
Including all solutions, [tex]\( t \)[/tex] can be written as:
[tex]\[
t = \frac{3 \ln(3)}{5} + \frac{2k\pi i}{5}
\][/tex]
for [tex]\( k \)[/tex] being any integer.
The complete solution is:
[tex]\[
t = \ln(3^{3/5}) - \frac{4\pi i}{5}
\][/tex]
