Write out all the written work that would be needed to solve this equation. solve for y. ln y = 2t +4

Sure! Here's the detailed written work to solve the equation \( \ln y = 2t + 4 \) for \( y \).

### Step 1: Write down the original equation.
\[
\ln y = 2t + 4
\]

### Step 2: Understand the logarithmic equation.
The equation \( \ln y = 2t + 4 \) means that the natural logarithm of \( y \) is equal to \( 2t + 4 \). The natural logarithm, denoted by \( \ln \), is the logarithm with base \( e \), where \( e \) is approximately 2.718.

### Step 3: Apply the exponential function to both sides.
To eliminate the natural logarithm, we exponentiate both sides of the equation. This step uses the fact that the exponential function \( e^x \) is the inverse of the natural logarithm \( \ln(x) \). So, apply the exponential function \( e^{(\cdot)} \) to both sides:

\[
e^{\ln y} = e^{2t + 4}
\]

### Step 4: Simplify the left side.
The exponential function and the natural logarithm are inverse functions, so \( e^{\ln y} \) simplifies to \( y \):

\[
y = e^{2t + 4}
\]

### Step 5: Write the final solution.
The solution for \( y \) is:

\[
y = e^{2t + 4}
\]

This is the final expression for \( y \) in terms of \( t \).

Would you like any further details or have any questions about this process? Here are five related questions to deepen your understanding:

1. How would you solve \( \ln(x) = 5y + 3 \) for \( x \)?
2. Can you solve the equation \( \ln(y - 3) = 2t + 4 \) for \( y \)?
3. What steps are needed to solve \( \ln(y) = 3t + 7 \) for \( y \) if there are additional coefficients or constants?
4. How do you differentiate the function \( y = e^{2t + 4} \) with respect to \( t \)?
5. Can you explain how logarithms are used to solve exponential equations?

**Tip:** When solving logarithmic equations, always remember the relationship between logarithms and exponents, as this is crucial for simplifying and solving the equations efficiently.

Learn how to solve the equation ln y = 2t + 4 for y with a detailed step-by-step solution. This problem involves logarithms and exponential functions, suitable for high school students.

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