Use logarithms to solve the equation for t. (Round your answer to two decimal places.)
30/(1+7 e^(-0.7 t))=10

We are given the equation:

\[
\frac{30}{1 + 7e^{-0.7t}} = 10
\]

We need to solve for \( t \) using logarithms. Here's the step-by-step solution:

### Step 1: Multiply both sides by \( 1 + 7e^{-0.7t} \) to eliminate the denominator.
\[
30 = 10(1 + 7e^{-0.7t})
\]

### Step 2: Expand the right-hand side.
\[
30 = 10 + 70e^{-0.7t}
\]

### Step 3: Subtract 10 from both sides.
\[
20 = 70e^{-0.7t}
\]

### Step 4: Divide both sides by 70 to isolate the exponential term.
\[
\frac{20}{70} = e^{-0.7t}
\]
\[
\frac{2}{7} = e^{-0.7t}
\]

### Step 5: Take the natural logarithm (ln) of both sides to eliminate the exponential.
\[
\ln\left(\frac{2}{7}\right) = -0.7t
\]

### Step 6: Solve for \( t \).
\[
t = \frac{\ln\left(\frac{2}{7}\right)}{-0.7}
\]

### Step 7: Compute the natural logarithm and divide.
\[
t = \frac{\ln(0.2857)}{-0.7} \approx \frac{-1.25276}{-0.7} \approx 1.79
\]

Thus, the solution is:

\[
t \approx 1.79
\]

Would you like further details or explanations? 

Here are 5 related questions:
1. How do logarithmic properties simplify solving exponential equations?
2. What is the importance of natural logarithms in solving exponential functions?
3. Can the solution change if we use a different base for the logarithm?
4. How do we interpret the value of \( t \) in real-world exponential models?
5. What happens if we have a different constant in the exponent?

**Tip:** Always check your work by substituting the solution back into the original equation to verify its correctness.

Use logarithms to solve the equation for t. (Round your answer to two decimal places.) 30/(1+7 e^(-0.7 t))=10

Learn how to solve the exponential equation 30/(1+7e^(-0.7t)) = 10 using logarithms. This step-by-step solution demonstrates the use of natural logarithms to solve for t, suitable for high school students in Grades 10-12. Key concepts include exponential functions and logarithmic properties.

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