How do you find the domain of $f(x) = \log_b(g(x))$?
1. Set g(x) > 0. 2. Solve for x. This gives the domain of f(x).
How do you solve for x in the equation $\log_b(x) = c$?
Rewrite the equation in exponential form: $x = b^c$.
How do you graph $y = a\log_b(x - h) + k$?
1. 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 $y = \log_b(x)$ 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 $y = \log_b(x)$ 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?
1. 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?
1. Replace f(x) with y. 2. Swap x and y. 3. Solve for y. 4. Replace y with $f^{-1}(x)$.
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.
What does the graph of $y = \log_b(x)$ with b > 1 tell us?
The function is increasing, and the graph is concave down.
What does the graph of $y = \log_b(x)$ 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 $b^y = x$.
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 $y = \log_b(x)$ relate to the graph of $y = b^x$?
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.
Explain the relationship between logarithmic and exponential functions.
Logarithmic functions are the inverses of exponential functions. They 'undo' each other.
How does the base 'b' affect the graph of $y = \log_b(x)$?
If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.
Why do logarithmic functions have a vertical asymptote at x = 0?
Because the inverse exponential function has a horizontal asymptote at y = 0, restricting the domain of the logarithm to positive values.
Describe the end behavior of a logarithmic function as x approaches infinity.
As x approaches infinity, y also approaches either positive or negative infinity, depending on the base and any transformations.
Explain how horizontal shifts affect the domain and asymptote of a logarithmic function.
A horizontal shift changes the vertical asymptote and, consequently, the domain. For example, $\log_b(x+k)$ shifts the asymptote to x = -k, and the domain becomes x > -k.
Why don't logarithmic functions have maximums, minimums, or inflection points?
Because they are always either increasing or decreasing and always concave up or concave down.
Explain the significance of the domain restriction for logarithmic functions.
The argument of a logarithm must be positive because you cannot raise a base to any power and get a non-positive result.
Describe the effect of a negative sign in front of a logarithmic function.
It reflects the graph across the x-axis, changing increasing functions to decreasing and vice versa, and affecting the end behavior.
How do transformations affect the range of a logarithmic function?
Vertical shifts and stretches can change the range but since the range is all real numbers, only vertical shifts change the graph.
Explain how to determine if a function is logarithmic based on its additive transformations.
If output values are proportional over equal input intervals, then the function is logarithmic.
