Problem 109 - Precalculus

Question

Determine whether each statement is true or false. The graphs of \(y=\log x\) and \(y=\ln x\) have the same \(x\) -intercept (1,0).

Step-by-Step Solution

Answer

True, both graphs intersect the x-axis at (1,0).

Step 1: Understanding Logarithms

The function \(y = \log x\) typically refers to the logarithm with base 10, known as the common logarithm. The function \(y = \ln x\) is the natural logarithm and has base \(e\). Both functions are undefined for \(x \leq 0\) and have a vertical asymptote at \(x = 0\).

Step 2: Identifying the x-intercepts

To find the \(x\)-intercept of a logarithmic function, we find the value of \(x\) when \(y = 0\). For \(y = \log x\) when \(y = 0\), solving \(\log x = 0\) gives \(x = 10^0 = 1\). Similarly, for \(y = \ln x\), solving \(\ln x = 0\) gives \(x = e^0 = 1\).

Step 3: Checking if Intercepts are Identical

Both \(y = \log x\) and \(y = \ln x\) have their \(x\)-intercepts at the point \((1, 0)\). Therefore, the statement asserts that both graphs intersect the \(x\)-axis at the same point.

Step 4: Determining Truth or Falsity

Since both functions have the \(x\)-intercept \((1, 0)\), the statement that their graphs have the same \(x\)-intercept is true.

Key Concepts

Natural Logarithm

The natural logarithm, represented as \(y = \ln x\), is a logarithmic function with base \(e\), which is approximately 2.71828. It is one of the most commonly used logarithms in mathematics, especially in calculus and exponential growth problems.

When you see \(\ln x\), think about what power you need to raise \(e\) to get \(x\). The natural logarithm has a few properties:

• \(\ln 1 = 0\), because \(e^0 = 1\).
• \(\ln e = 1\), because \(e^1 = e\).
• The graph of \(\ln x\) passes through the point \((1, 0)\), meaning its \(x\)-intercept is at \(x=1\).
• This graph is defined only for \(x > 0\).

Natural logarithms arise naturally when dealing with continuous compounding, certain differential equations, and when calculating things like time or decay in statistics and finance.

Common Logarithm

The common logarithm is denoted by \(y = \log x\) and uses 10 as its base. It is widely used in various scientific disciplines such as chemistry and physics, as well as in logarithmic scaling and sound intensity calculations.

The common logarithm is helpful because it simplifies multiplication into addition, especially when working with powers of 10. Here are some basic properties:

• \(\log 1 = 0\), because \(10^0 = 1\).
• \(\log 10 = 1\), since \(10^1 = 10\).
• The graph of \(\log x\) also has its \(x\)-intercept at \((1, 0)\).
• This function is defined only for \(x > 0\) as well.

In everyday applications, when we express very large or small numbers, common logarithms simplify calculations by working with their orders of magnitude.

Vertical Asymptote

A vertical asymptote occurs in a graph where the function approaches a line but never actually touches it. For the graphs of both the natural logarithm \(\ln x\) and the common logarithm \(\log x\), the vertical asymptote is at \(x = 0\).

Why does this happen? Since you can't take the logarithm of zero or a negative number (as it is undefined), the values "approach" tangentially and indefinitely near zero, but never actually reach or pass it. Here are a few characteristics of vertical asymptotes in logarithmic functions:

• The graph will sharply rise as \(x\) approaches zero from the right.
• The value never decreases back to allow for \(x\leq 0\).
• It indicates the boundary or restriction of the domain, focusing only on \(x > 0\).

Understanding vertical asymptotes helps in recognizing the behavior of functions near undefined values, especially when plotting the graphs of those functions.