Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms.​ Then, use a calculator to obtain a decimal approximation for the solution.
e Superscript 1 minus 7 x Baseline equals 323
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Part 1
The solution set expressed in terms of logarithms is StartSet nothing EndSet .

​(Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the expression. Use ln for natural logarithm and log for common​ logarithm.)

Given the exponential equation \( e^{1 - 7x} = 323 \), we need to express the solution set in terms of natural logarithms or common logarithms and then find a decimal approximation.

### Step 1: Isolate the exponential term
We start by isolating the exponential term:

\[
e^{1 - 7x} = 323
\]

### Step 2: Take the natural logarithm of both sides
To solve for \( x \), take the natural logarithm (ln) of both sides:

\[
\ln\left(e^{1 - 7x}\right) = \ln(323)
\]

### Step 3: Simplify using the properties of logarithms
Using the property \( \ln(e^a) = a \), we can simplify the left side:

\[
1 - 7x = \ln(323)
\]

### Step 4: Solve for \( x \)
Now, solve for \( x \):

\[
-7x = \ln(323) - 1
\]

\[
x = \frac{1 - \ln(323)}{7}
\]

So, the solution set expressed in terms of natural logarithms is:

\[
\left\{ \frac{1 - \ln(323)}{7} \right\}
\]

### Step 5: Decimal approximation
To find the decimal approximation, calculate the value:

\[
x \approx \frac{1 - \ln(323)}{7}
\]

Using a calculator:

\[
\ln(323) \approx 5.777
\]

\[
x \approx \frac{1 - 5.777}{7} \approx \frac{-4.777}{7} \approx -0.6824
\]

So, the decimal approximation is approximately \( x \approx -0.6824 \).

### Final Answer:
The solution set in logarithmic form is:

\[
\left\{ \frac{1 - \ln(323)}{7} \right\}
\]

And the decimal approximation for the solution is:

\[
x \approx -0.6824
\]

Would you like further details or have any questions? Here are some related questions to consider:

1. How do you solve other exponential equations using logarithms?
2. What is the difference between natural logarithms and common logarithms?
3. How do you apply logarithm properties to simplify complex expressions?
4. Can you convert a logarithmic equation back to an exponential form?
5. How can you use a calculator to evaluate logarithmic expressions?

**Tip:** Remember that logarithms are the inverse operations of exponentiation, which is a crucial concept in solving exponential equations.

Learn how to solve the exponential equation e^(1 - 7x) = 323 using logarithms and find its decimal approximation. This problem covers exponential equations, natural logarithms, and provides a step-by-step solution. Suitable for high school students in Grades 10-12.

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