Solve the logarithmic equation algebraically. Approximate the result to three decimal places. (If there is no solution, enter NO SOLUTION.)
2.2 = ln 4x

To solve the equation \(2.2 = \ln(4x)\) algebraically, we will follow these steps:

### Step 1: Rewrite the Equation in Exponential Form
Since we have a natural logarithm, we can rewrite the equation as:
\[
\ln(4x) = 2.2
\]
Using the property that if \(\ln(a) = b\), then \(a = e^b\), we rewrite this as:
\[
4x = e^{2.2}
\]

### Step 2: Solve for \(x\)
Now we divide both sides by 4 to isolate \(x\):
\[
x = \frac{e^{2.2}}{4}
\]

### Step 3: Calculate \(e^{2.2}\)
Using the approximation \(e^{2.2} \approx 9.025\), we get:
\[
x = \frac{9.025}{4} \approx 2.256
\]

### Final Answer
The approximate solution to the equation, rounded to three decimal places, is:
\[
x \approx 2.256
\]

Let me know if you'd like more details or have any questions.

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Here are five related questions:
1. How would the solution change if the base of the logarithm were different?
2. How do you solve if \(x\) is in a different position within the logarithmic argument?
3. What is the effect of adding constants to both sides of a logarithmic equation?
4. How do you check if a solution satisfies a logarithmic equation?
5. How can logarithmic equations with different bases be solved?

**Tip:** Remember that you can always rewrite logarithmic equations in exponential form to simplify solving.

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. (If there is no solution, enter NO SOLUTION.) 2.2 = ln 4x

Learn how to solve the logarithmic equation 2.2 = ln(4x) with a detailed, step-by-step algebraic solution. This problem explores key logarithmic properties, using exponential functions to approximate the solution to three decimal places. Suitable for high school students in Grades 10-12.

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