Answer:
Explanation:
The equation \(8e^x - 1 = 0\) can be solved to find the value of \(x\).
To solve for \(x\), we need to isolate the exponential term, \(e^x\).
Here are the steps to solve the equation:
1. Add 1 to both sides of the equation to isolate the exponential term:
\(8e^x = 1\)
2. Divide both sides of the equation by 8:
\(\frac{{8e^x}}{8} = \frac{1}{8}\)
3. Simplify:
\(e^x = \frac{1}{8}\)
4. To solve for \(x\), we can take the natural logarithm (ln) of both sides of the equation:
\(\ln(e^x) = \ln\left(\frac{1}{8}\right)\)
5. Since \(\ln(e^x)\) and \(e^x\) are inverse functions, they cancel each other out:
\(x = \ln\left(\frac{1}{8}\right)\)
6. Use the properties of logarithms to simplify further:
\(x = \ln(1) - \ln(8)\)
7. Simplify:
\(x = -\ln(8)\)
Therefore, the solution for \(x\) in the equation \(8e^x - 1 = 0\) is \(x = -\ln(8)\).