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------------------------------------------------ Solve for [tex]t[/tex].

[tex]e^{t} = 156[/tex]

To solve the equation [tex]\( e^t = 156 \)[/tex] for [tex]\( t \)[/tex], we can use the natural logarithm. Natural logarithm, denoted by [tex]\( \ln \)[/tex], is particularly useful when dealing with exponential equations involving the base [tex]\( e \)[/tex].

Here's how to solve it step-by-step:

1. Understand the Equation: We start with the equation [tex]\( e^t = 156 \)[/tex]. Our goal is to isolate [tex]\( t \)[/tex].

2. Take the Natural Logarithm of Both Sides: To solve for [tex]\( t \)[/tex], take the natural logarithm of both sides of the equation. This step is based on the property that [tex]\( \ln(e^t) = t \)[/tex].

[tex]\[
\ln(e^t) = \ln(156)
\][/tex]

3. Simplify Using Logarithm Rules: By the logarithm rule [tex]\( \ln(e^t) = t \cdot \ln(e) \)[/tex] and knowing that [tex]\( \ln(e) = 1 \)[/tex], we simplify the left side:

[tex]\[
t \cdot 1 = \ln(156)
\][/tex]

So, this simplifies to:

[tex]\[
t = \ln(156)
\][/tex]

4. Calculate the Logarithm: Finally, calculate [tex]\( \ln(156) \)[/tex].

The result is approximately:

[tex]\[
t \approx 5.049856007249537
\][/tex]

Therefore, the solution to the equation [tex]\( e^t = 156 \)[/tex] is approximately [tex]\( t = 5.0499 \)[/tex].