Understanding Logarithmic Functions and Transformations

The video tutorial explains the domain, range, and vertical asymptote of the function g(x) = -ln(x-6). It begins with a review of basic logarithmic functions, their properties, and how they relate to exponential functions. The domain of g(x) is determined by solving x-6 > 0, resulting in x > 6. The vertical asymptote is found at x = 6. The range remains all real numbers despite the negative sign in the function. The tutorial also covers transformations of the basic natural log function, including reflection across the x-axis and horizontal shifts. Graphical representations are used to verify these transformations and properties.

What is the general form of a logarithmic function?

• y = x^b

• y = ln(x)

• y = b^x

• y = log base b of x

Which of the following is a property of the basic natural log function?

• It has a vertical asymptote at x = 1

• It has a horizontal asymptote at y = 0

• Its range is all real numbers

• It is undefined for x > 0

What is the horizontal intercept of the basic natural log function?

• (1, 1)

• (0, 0)

• (1, 0)

• (0, 1)

For the function g(x) = -ln(x-6), what is the domain?

• x > 0

• x > 6

• x < 6

• x < 0

What is the vertical asymptote of the function g(x) = -ln(x-6)?

• x = 1

• x = -6

• x = 6

• x = 0

How does the negative sign in g(x) = -ln(x-6) affect the graph?

• It stretches the graph vertically

• It shifts the graph left

• It reflects the graph across the x-axis

• It shifts the graph up

What transformation occurs when x is replaced by x-6 in the function g(x) = -ln(x-6)?

• Vertical shift up

• Vertical shift down

• Horizontal shift right

• Horizontal shift left

How does the graph of y = -ln(x) differ from y = ln(x)?

• It is shifted down

• It is shifted up

• It is reflected across the y-axis

• It is reflected across the x-axis

In the graph of g(x) = -ln(x-6), what happens as x approaches 6 from the right?

• The function values approach zero

• The function values approach negative infinity

• The function values approach positive infinity

• The function values remain constant

What is the range of the function g(x) = -ln(x-6)?

• All real numbers

• x > 0

• x < 0

• x > 6