e^{9x}=8 what is the approximate solution to four decimals

To solve the equation \( e^{9x} = 8 \), we can take the natural logarithm of both sides:

\[
\ln(e^{9x}) = \ln(8)
\]

Using the property of logarithms, this simplifies to:

\[
9x = \ln(8)
\]

Now, we solve for \( x \):

\[
x = \frac{\ln(8)}{9}
\]

Calculating \( \ln(8) \):

\[
\ln(8) \approx 2.0794
\]

Now divide by 9:

\[
x \approx \frac{2.0794}{9} \approx 0.2310
\]

Thus, the approximate solution is:

\[
\boxed{0.2310}
\]

Would you like more details on this process or have any questions? Here are some related questions you might consider:

1. How do logarithms help in solving exponential equations?
2. What is the value of \( \ln(2) \) and how is it useful?
3. Can you explain the properties of logarithms?
4. How would you solve a similar equation, such as \( e^{5x} = 15 \)?
5. What are some applications of exponential equations in real life?

**Tip:** Always check your solution by substituting it back into the original equation!

e^{9x} = 8 what is the approximate solution to four decimals

Find the approximate solution to the exponential equation e^{9x} = 8 to four decimal places. This guide covers the use of logarithmic properties to simplify and solve for x, making it suitable for high school math students. Key concepts include exponential functions and logarithmic transformation.

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