Solve for the value of  x
  at which the following two curves intersect.

 

y=ln(6x+8)
 
 

y=ln(x)+7
 


To find the value of \( x \) at which the two curves intersect, we need to set the equations equal to each other and solve for \( x \). 

The two given equations are:

1. \( y = \ln(6x + 8) \)
2. \( y = \ln(x) + 7 \)

### Step 1: Set the equations equal to each other

\[
\ln(6x + 8) = \ln(x) + 7
\]

### Step 2: Isolate the logarithmic terms

Subtract \( \ln(x) \) from both sides:

\[
\ln(6x + 8) - \ln(x) = 7
\]

### Step 3: Apply the logarithmic property

Use the logarithmic property \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \):

\[
\ln\left(\frac{6x + 8}{x}\right) = 7
\]

### Step 4: Exponentiate both sides

Raise both sides to the power of \( e \) to cancel the natural logarithm:

\[
\frac{6x + 8}{x} = e^7
\]

### Step 5: Simplify the equation

Multiply both sides by \( x \):

\[
6x + 8 = x e^7
\]

### Step 6: Collect like terms

Move all terms involving \( x \) to one side:

\[
6x - x e^7 = -8
\]

Factor out \( x \):

\[
x(6 - e^7) = -8
\]

### Step 7: Solve for \( x \)

Divide both sides by \( (6 - e^7) \):

\[
x = \frac{-8}{6 - e^7}
\]

This is the solution for \( x \), which gives the value where the two curves intersect.

Would you like further details or clarification on this? 

Here are 5 related questions to expand your understanding:

1. How can we check if the value of \( x \) is valid for both functions?
2. What are the properties of logarithmic functions that make them suitable for solving equations like this?
3. What does the intersection of these two curves represent geometrically?
4. How would this process change if one equation used a base other than \( e \) for the logarithm?
5. Could there be more than one intersection point for different logarithmic curves? Why or why not?

**Tip:** Always remember to check the domain restrictions of logarithmic functions, as they require their arguments to be positive.

Solve for the value of x at which the following two curves intersect: y = ln(6x + 8) and y = ln(x) + 7.

Learn how to find the intersection point of two logarithmic functions, y = ln(6x + 8) and y = ln(x) + 7. This step-by-step guide covers logarithmic properties, exponentiation, and algebraic manipulation. Suitable for advanced high school students in Grades 11-12.

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