To solve for \(t\) in the equation \(50 = 100 \cdot 0.945^t\), follow these steps:

### Step 1: Isolate the exponential term
Divide both sides by 100:
\[
\frac{50}{100} = 0.945^t
\]
\[
0.5 = 0.945^t
\]

### Step 2: Take the natural logarithm of both sides
Applying the natural logarithm \(\ln\) to both sides:
\[
\ln(0.5) = \ln(0.945^t)
\]

### Step 3: Apply the logarithm power rule
\(\ln(a^b) = b \cdot \ln(a)\):
\[
\ln(0.5) = t \cdot \ln(0.945)
\]

### Step 4: Solve for \(t\)
Divide both sides by \(\ln(0.945)\):
\[
t = \frac{\ln(0.5)}{\ln(0.945)}
\]

### Step 5: Calculate the values
Using approximations:
\[
\ln(0.5) \approx -0.6931, \quad \ln(0.945) \approx -0.0568
\]

Now divide:
\[
t = \frac{-0.6931}{-0.0568} \approx 12.2
\]

### Final Answer:
\[
t \approx 12.2
\]

Would you like me to go over any step or expand on the method?

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Here are 5 related questions for practice:
1. How do you solve for \(t\) in \(P = P_0 \cdot (1 + r)^t\)?
2. What is the general form of exponential decay equations?
3. How does the base of the logarithm affect calculations?
4. What does \(t\) represent in exponential models for decay and growth?
5. How do natural logs simplify solving exponential equations?

**Tip:** If the base isn't specified in the problem, using the natural logarithm \(\ln\) makes calculations easier!

Learn how to solve the exponential decay equation 50 = 100 * 0.945^t for t. This step-by-step guide explores the use of logarithms and exponential decay in algebra, suitable for high school students in Grades 10-12.

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