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What is the general form of a logarithmic function?
$y = \log_b(x)$
What is the limit of $a\log_b(x)$ as x approaches 0 from the right?
$\lim_{x \to 0^+} a\log_b(x) = \pm \infty$
What is the limit of $a\log_b(x)$ as x approaches infinity?
$\lim_{x \to \infty} a\log_b(x) = \pm \infty$
How do you represent a horizontal shift of a logarithmic function?
$g(x) = \log_b(x + k)$
If $y = \log_b(x)$, what is the equivalent exponential form?
$b^y = x$
What is the logarithmic identity for $\log_b(1)$?
$\log_b(1) = 0$
What is the logarithmic identity for $\log_b(b)$?
$\log_b(b) = 1$
What is the change of base formula?
$\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$
What is the product rule for logarithms?
$\log_b(MN) = \log_b(M) + \log_b(N)$
What is the quotient rule for logarithms?
$\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$
What is a logarithmic function?
The inverse of an exponential function; 'undoes' exponentiation.
What is the domain of $y = \log_b(x)$?
x > 0 (positive real numbers)
What is the range of $y = \log_b(x)$?
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 $\log_b(x)$. Must be positive.
What is the base of a logarithm?
The value 'b' in $\log_b(x)$. 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.
What are the differences between exponential growth and logarithmic growth?
Exponential: Rapid increase, unbounded | Logarithmic: Slow increase, bounded by asymptote.
What are the differences between $y = \log_b(x)$ and $y = -\log_b(x)$?
$\log_b(x)$: Increasing (if b > 1), positive y-values for x > 1 | $-\log_b(x)$: Decreasing (if b > 1), negative y-values for x > 1.
What are the differences between horizontal and vertical shifts of $y = \log_b(x)$?
Horizontal: Changes the domain and asymptote | Vertical: Changes the range (though range is all real numbers).
What are the differences between $\log(x*y)$ and $\log(x+y)$?
$\log(x*y)$: Can be expanded to $\log(x) + \log(y)$ | $\log(x+y)$: Cannot be simplified further.
What are the differences between $\log_b(x)$ where b > 1 and 0 < b < 1?
b > 1: Increasing function | 0 < b < 1: Decreasing function.
What are the differences between the domain and range of exponential and logarithmic functions?
Exponential: Domain is all real numbers, range is y > 0 | Logarithmic: Domain is x > 0, range is all real numbers.
What are the differences between the graphs of $y = \log_b(x)$ and $y = b^x$?
$\log_b(x)$: Vertical asymptote at x = 0 | $b^x$: Horizontal asymptote at y = 0. They are reflections across y = x.
What are the differences between solving logarithmic and exponential equations?
Logarithmic: Often involves combining logs and converting to exponential form | Exponential: Often involves isolating the exponential term and taking the logarithm of both sides.
What are the differences between the effects of vertical stretches and compressions on logarithmic functions?
Vertical Stretch: Makes the graph steeper | Vertical Compression: Makes the graph less steep.
What are the differences between the product rule and quotient rule for logarithms?
Product Rule: $\log_b(MN) = \log_b(M) + \log_b(N)$ | Quotient Rule: $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$