Problem 53 - College Algebra

Question

For the following exercises, use a graphing calculator to find approximate solutions to each equation.$$\log (x-1)+2=\ln (x-1)+2$$

Step-by-Step Solution

Answer

The approximate solution is when \( x = 1.374 \).

Step 1: Understanding the Equation

The given equation is \( \log (x-1) + 2 = \ln (x-1) + 2 \). This simplifies to \( \log (x-1) = \ln (x-1) \) since adding 2 on both sides does not change the equality. Here, \( \log \) represents the logarithm with base 10, and \( \ln \) represents the natural logarithm with base \( e \).

Step 2: Simplifying the Equation

Remove the constant \( +2 \) from both sides, involving only the logarithmic expressions. We now have \( \log (x-1) = \ln (x-1) \). This implies that the functions must intersect at some point on the graph.

Step 3: Graphing the Functions

Graph \( y = \log(x-1) \) and \( y = \ln(x-1) \) using a graphing calculator. You will look for the point of intersection which represents the value of \( x \) that solves the equation.

Step 4: Identifying the Intersection

Using the graph on the calculator, find the point where the two curves \( y = \log(x-1) \) and \( y = \ln(x-1) \) intersect. This point of intersection will give the value of \( x - 1 \).

Step 5: Calculating the Solution

Once the intersection coordinate \( x - 1 \) is determined, add 1 to this value to solve for \( x \). This will give the solution of the original equation.

Key Concepts

Logarithmic Equations

Logarithmic equations involve expressions with logarithms, which are the inverses of exponential functions. In the equation \( \log(x-1) = \ln(x-1) \), both terms contain a logarithmic function, but with different bases.

• \( \log \) generally implies a base of 10 (common logarithm).
• \( \ln \) uses a base \( e \) (natural logarithm), where \( e \approx 2.718 \).

Understanding the base is crucial. It affects how the equations behave and where they intersect. We aim to find an \( x \)-value for which both logarithms equal the same number, meaning \( \log(x-1) = \ln(x-1) \).
To solve, simplify steps to isolate the logarithmic terms and use tools like graphing calculators to visualize and solve equations for approximations.

Natural Logarithm

The natural logarithm, denoted \( \ln \), is a logarithm with base \( e \), a mathematical constant roughly equal to 2.718. This form of logarithm frequently appears in calculus and natural growth phenomena.

• Used to model exponential growth or decay, such as populations or radioactive decay.
• Natural logarithms are helpful in solving exponential equations where the base is \( e \).

In the context of our exercise, comparing the natural logarithm \( \ln(x-1) \) with the common logarithm \( \log(x-1) \) helps find their point of intersection. This relationship is key in understanding how different logarithmic forms compare directly on a graph.
Exploring properties like the change of base formula may also aid in analyzing these forms algebraically, either for approximation or simplification purposes.

Intersection of Graphs

When we talk about the intersection of graphs, especially in the context of equations like this, we refer to the point where two different functions meet on a graph. In the exercise, finding where \( y = \log(x-1) \) and \( y = \ln(x-1) \) intersect is our main goal.

• Plot both functions on a graphing calculator.
• Look for the x-coordinate where they intersect; this location represents a point where both functions have the same output.

This point of intersection is crucial because it tells us the value of \( x - 1 \) that satisfies the original equation \( \log(x-1) = \ln(x-1) \).
By interpreting this graphically, not only do we solve the equation, but we also visually understand how different logarithmic bases come together. This visualization aids in grasping the relationship between non-identical log functions and their intersections in a straightforward way that calculations alone may not completely elucidate.