All Flashcards
What does the graph of with b > 1 tell us?
The function is increasing, and the graph is concave down.
What does the graph of with 0 < b < 1 tell us?
The function is decreasing, and the graph is concave down.
How does a vertical asymptote appear on the graph of a logarithmic function?
As a vertical line that the graph approaches but never crosses, indicating a domain restriction.
How does a horizontal shift affect the graph of a logarithmic function?
It moves the entire graph left or right, changing the position of the vertical asymptote.
What does a reflection across the x-axis do to the graph of a logarithmic function?
It inverts the function, changing increasing functions to decreasing and vice versa.
How can you identify the base of a logarithmic function from its graph?
Look for a point (x, y) on the graph and solve for b in the equation .
What does the steepness of a logarithmic graph indicate?
It indicates the rate of change of the function. Steeper graphs have a faster rate of change near the asymptote.
How does the graph of relate to the graph of ?
They are reflections of each other across the line y = x.
What does a vertical stretch of a logarithmic function look like on its graph?
The graph appears to be stretched vertically away from the x-axis.
How can you determine the domain of a transformed logarithmic function from its graph?
Identify the vertical asymptote; the domain is all x-values greater than (or less than, depending on reflection) the asymptote's x-value.
What is a logarithmic function?
The inverse of an exponential function; 'undoes' exponentiation.
What is the domain of ?
x > 0 (positive real numbers)
What is the range of ?
All real numbers
What is a vertical asymptote?
A vertical line that the graph of a function approaches but never touches.
What is the argument of a logarithm?
The value inside the logarithm, e.g., 'x' in . Must be positive.
What is the base of a logarithm?
The value 'b' in . Determines if the function is increasing or decreasing.
What does concavity mean for a logarithmic function?
Describes the curve's shape: either concave up or concave down, but not both.
What is horizontal shift?
A transformation of a graph where the entire graph is moved left or right.
Define end behavior.
Describes how the function behaves as x approaches positive or negative infinity.
What are transformations of functions?
Changes to a function's graph, such as shifts, stretches, or reflections.
How do you find the domain of ?
- Set g(x) > 0. 2. Solve for x. This gives the domain of f(x).
How do you solve for x in the equation ?
Rewrite the equation in exponential form: .
How do you graph ?
- Identify the vertical asymptote at x = h. 2. Plot a few key points. 3. Sketch the curve, considering the base 'b' and the vertical stretch 'a'.
How do you determine the end behavior of as x approaches 0?
Consider the base 'b'. If b > 1, y approaches negative infinity. If 0 < b < 1, y approaches positive infinity.
How do you determine the end behavior of as x approaches infinity?
Consider the base 'b'. If b > 1, y approaches positive infinity. If 0 < b < 1, y approaches negative infinity.
How do you solve logarithmic equations with multiple logarithms?
- Combine logarithms using properties. 2. Convert to exponential form. 3. Solve for x. 4. Check for extraneous solutions.
How do you find the inverse of a logarithmic function?
- Replace f(x) with y. 2. Swap x and y. 3. Solve for y. 4. Replace y with .
How do you apply transformations to a logarithmic function?
Apply transformations in the correct order: horizontal shifts, stretches/compressions, reflections, and vertical shifts.
How do you check if a solution to a logarithmic equation is extraneous?
Substitute the solution back into the original equation and ensure the argument of each logarithm is positive.
How do you find the x-intercept of a logarithmic function?
Set y = 0 and solve for x.