Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

Is there a real number \(x\) such that \(x=\log x\) ? Decide by displaying graphically the system $$ \left\\{\begin{array}{l} y=x \\ y=\log x \end{array}\right. $$

Step-by-Step Solution

Answer

No, there is no real number where \( x = \log x \).

Step 1: Understanding the Functions

The two functions given are \( y = x \) and \( y = \log x \). The function \( y = x \) is a straight line passing through the origin with a slope of 1. The function \( y = \log x \) is the logarithmic curve for base 10 (commonly natural log is used, but since base isn't given, assume common log). It is defined for \( x > 0 \) and it passes through the point (1,0).

Step 2: Plotting the Graph of \( y = x \)

The graph of \( y = x \) is a straight line. For every increment in \( x \), the value of \( y \) also increments by the same amount. To plot this graph, take some values like when \( x = 0 \), \( y = 0 \); when \( x = 1 \), \( y = 1 \); and when \( x = 2 \), \( y = 2 \). Connect these points to form a straight line.

Step 3: Plotting the Graph of \( y = \log x \)

The graph of \( y = \log x \) is a curve that approaches negative infinity as \( x \) approaches zero from the right, and it gradually increases as \( x \) increases, crossing the point (1,0). To plot this graph, find some values like when \( x = 1 \), \( y = \log 1 = 0 \); when \( x = 10 \), \( y = \log 10 = 1 \); and when \( x = 0.1 \), \( y = \log 0.1 = -1 \). Connect these points to form the logarithmic curve.

Step 4: Analyzing Intersection Points

With both graphs plotted on the same axes, look for intersection points. The point \( (x, y) \) where the graphs intersect is a solution to the equation \( x = \log x \). Check if there's any real \( x \) such that both functions share the same value at the same \( x \)-coordinate.

Step 5: Conclusion based on Graphs

Upon analyzing the plotted graphs, you would see there is no real point of intersection between \( y = x \) and \( y = \log x \). The graph \( y = x \) cuts through the curve but never intersects in positive \( x \)-axis as \( \log x \). Hence, there is no real number solution for the equation \( x = \log x \).

Key Concepts

Graphing Functions

When tackling problems involving equations like \( y = x \) and \( y = \log x \), visualizing these functions on a graph can be incredibly helpful. For \( y = x \), we have a simple linear function. It forms a straight diagonal line through the origin, representing a slope of 1. This means for every unit increase in \( x \), \( y \) increases by the same unit. On the other hand, \( y = \log x \) represents a logarithmic function, which forms a curve on the graph. This curve is only defined for \( x > 0 \). It starts from negative infinity when \( x \) is close to zero and gradually increases. It crosses the \( x \)-axis precisely at the point (1,0), indicating that the logarithm of 1 is zero.Plotting these functions on the same set of axes provides a clear visual contrast between the steady progression of the linear function and the slower, creeping rise of the logarithmic function.

System of Equations

A system of equations involves finding the common solutions to multiple equations. Here, the goal is to find an \( x \) value that satisfies both \( y = x \) and \( y = \log x \) simultaneously. Graphically, this can be interpreted as finding the points where the two graphs intersect. If a system has a solution, there will be at least one intersection point where the graphs meet. However, if there are no such points, the system lacks a solution.The challenge with this problem lies in the nature of the functions themselves. The linear nature of \( y = x \) contrasts with the logarithmic nature of \( y = \log x \), making it more complex to identify intersections just by calculation. Hence, visual analysis is a powerful tool here.

Intersection Points

Intersection points occur where the values of the functions are equal at a particular \( x \) value. In practical terms, it translates to the point or points where two graphs cross paths on a graph. In this problem, determining intersection points involves checking where the line \( y = x \) meets the curve \( y = \log x \). By examining the visual graph, we aim to find \( x \) values common to both equations. However, after plotting the graphs, it becomes clear that \( y = x \) and \( y = \log x \) do not intersect for any real number. The line and the curve approach each other but never truly meet along the real x-axis. Therefore, the conclusion - there is no real solution for \( x = \log x \) - is apparent through graph analysis.