To solve the equation [tex]\(9e^x = 19\)[/tex] for [tex]\(x\)[/tex], follow these steps:
### Step 1: Isolate [tex]\(e^x\)[/tex]
First, we need to isolate the term [tex]\(e^x\)[/tex] on one side of the equation. We can do this by dividing both sides by 9:
[tex]\[
e^x = \frac{19}{9}
\][/tex]
### Step 2: Apply the Natural Logarithm
Next, we take the natural logarithm (denoted as [tex]\(\ln\)[/tex]) of both sides of the equation. The natural logarithm is the inverse function of the exponential function [tex]\(e^x\)[/tex], which will allow us to solve for [tex]\(x\)[/tex]:
[tex]\[
\ln(e^x) = \ln\left(\frac{19}{9}\right)
\][/tex]
### Step 3: Simplify the Left Side
Using the property of logarithms [tex]\(\ln(e^x) = x\cdot\ln(e)\)[/tex] and knowing that [tex]\(\ln(e) = 1\)[/tex], the left side simplifies to:
[tex]\[
x = \ln\left(\frac{19}{9}\right)
\][/tex]
### Step 4: Calculate the Natural Logarithm
To find the value of [tex]\(x\)[/tex], calculate the natural logarithm of [tex]\(\frac{19}{9}\)[/tex]:
[tex]\[
x = \ln\left(\frac{19}{9}\right) \approx 0.7472144018302211
\][/tex]
### Conclusion
The solution to the equation [tex]\(9e^x = 19\)[/tex] for [tex]\(x\)[/tex] is approximately:
[tex]\[
x \approx 0.7472144018302211
\][/tex]
