Equations of Vertical Asymptotes for Logarithmic Functions

TLDR; Explaining the equations of vertical asymptotes for logarithmic functions, including domains and graphical representations.

⚖️ Equation of Vertical Asymptote

The speaker explains how to find the equation of the vertical asymptote for a logarithmic function.

For the function f(x) = log(x - 4), it's noted that the base is not given, indicating a base 10 or common log.

The exponential equation 10 raised to the power of y = (x - 4) is equivalent to the logarithmic function.

To find the domain, the inequality x - 4 > 0 is solved, resulting in the domain x > 4.

The equation of the vertical asymptote is found by solving x - 4 = 0, leading to x = 4.

📈 Graphical Representation

The graphical representation of the logarithmic function f(x) = log(x - 4) is discussed.

By selecting values of y and solving for x, the graph approaches but never reaches x = 4, indicating a vertical asymptote.

It's emphasized that the graph never touches the vertical line x = 4, leading to the domain x > 4.

🔢 Exploring the 2nd Function

The second exponential function, f(x) = natural log of (x + 1), is introduced.

It's highlighted that natural log is log base e, and the corresponding exponential form is e raised to the power of y = (x + 1).

The domain is determined by solving the inequality x + 1 > 0, resulting in the domain x > -1.

The equation of the vertical asymptote is found by solving x + 1 = 0, leading to x = -1.

📊 Verification through Graphs

The graphical representation of the natural log function f(x) = natural log of (x + 1) is discussed.

The graph approaches but never touches the vertical line x = -1, indicating the vertical asymptote, and the domain is confirmed to be x > -1.

📚 Summing Up

The explanations and graphical representations are summarized for better understanding the domains and equations of the vertical asymptotes.

The hope is expressed that the explanations and graphs aid in comprehending these concepts.