Solve for y in terms of x:

ln(2y−5)+ln(2)=5x+ln(5x)

*Simplify your answer to the point where the natural logarithm function ln does not appear in your answer.*

To solve the equation for \( y \) in terms of \( x \), we start with the given equation:

\[ \ln(2y - 5) + \ln(2) = 5x + \ln(5x) \]

### Step 1: Use the properties of logarithms

First, use the property of logarithms that states \(\ln(a) + \ln(b) = \ln(ab)\):

\[ \ln[(2y - 5) \cdot 2] = 5x + \ln(5x) \]

This simplifies to:

\[ \ln(4y - 10) = 5x + \ln(5x) \]

### Step 2: Exponentiate both sides to remove the natural logarithm

Exponentiate both sides with base \( e \):

\[ e^{\ln(4y - 10)} = e^{5x + \ln(5x)} \]

Since \( e^{\ln(a)} = a \), we have:

\[ 4y - 10 = e^{5x} \cdot 5x \]

### Step 3: Solve for \( y \)

Isolate \( y \) by adding 10 to both sides:

\[ 4y = 5x e^{5x} + 10 \]

Then divide by 4:

\[ y = \frac{5x e^{5x} + 10}{4} \]

### Final Answer

The solution for \( y \) in terms of \( x \) is:

\[ y = \frac{5x e^{5x} + 10}{4} \]

This is the simplified form where the natural logarithm function \( \ln \) does not appear.

Would you like further details or have any questions? Here are five related questions:

1. What are the properties of logarithms and how are they used in simplifying equations?
2. How does exponentiation help in solving logarithmic equations?
3. What is the significance of isolating \( y \) in terms of \( x \)?
4. Can we solve for \( x \) in terms of \( y \) from the same equation?
5. How can we check if the solution \( y = \frac{5x e^{5x} + 10}{4} \) is correct?

**Tip:** When dealing with logarithmic equations, remember to apply the properties of logarithms to combine or separate terms before using exponentiation to simplify.

Solve for y in terms of x: ln(2y−5) + ln(2) = 5x + ln(5x). Simplify your answer to the point where the natural logarithm function ln does not appear in your answer.

This page explains how to solve the logarithmic equation ln(2y−5) + ln(2) = 5x + ln(5x), with a step-by-step solution that includes logarithmic properties and exponentiation. Learn how to simplify the equation and express y in terms of x without the natural logarithm. Suitable for Grades 11-12.

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