Solve the exponential equation algebraically. Approximate the result to three decimal places. (Enter your answers as a comma-separated list.) 7 6 − *x* = 219

To solve the exponential equation \( 7^{6 - x} = 219 \) algebraically, we can proceed with the following steps:

1. **Rewrite the equation in logarithmic form:** 
   \[
   7^{6 - x} = 219
   \]
   Take the logarithm of both sides (using the natural logarithm, \( \ln \), or common logarithm, \( \log \), works equally well).

2. **Apply logarithms:**  
   \[
   \ln(7^{6 - x}) = \ln(219)
   \]

3. **Use the power rule of logarithms:**  
   \[
   (6 - x) \ln(7) = \ln(219)
   \]

4. **Isolate \( x \):**
   \[
   6 - x = \frac{\ln(219)}{\ln(7)}
   \]
   \[
   x = 6 - \frac{\ln(219)}{\ln(7)}
   \]

5. **Calculate the value using a calculator:**  
   First, find \( \ln(219) \approx 5.389 \) and \( \ln(7) \approx 1.946 \).

   Substitute these values:
   \[
   x = 6 - \frac{5.389}{1.946}
   \]
   \[
   x = 6 - 2.769 \approx 3.231
   \]

So, the approximate solution to three decimal places is:
\[
x \approx 3.231
\]

Would you like further details, or do you have any questions?

Here are some related questions you might find helpful:
1. How do logarithmic properties apply in solving exponential equations?
2. Can this equation be solved using common logarithms instead of natural logarithms?
3. What is the impact of changing the base of an exponent in an exponential equation?
4. Why is it useful to approximate logarithmic values in calculations?
5. How can exponential equations be solved using graphing instead of algebraic methods?

**Tip:** Remember that logarithmic functions are helpful in bringing exponents down as coefficients, simplifying exponential equations significantly.

Solve the exponential equation algebraically. Approximate the result to three decimal places. (Enter your answers as a comma-separated list.) 7 6 − *x* = 219

Learn how to solve the exponential equation 7^(6 - x) = 219 using logarithms. Step-by-step guide to applying logarithmic properties and calculating the result. This problem is suitable for students in Grades 9-11 studying algebra and exponential equations.

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