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What are the differences between exponential and logarithmic functions regarding asymptotes?
Exponential: Horizontal asymptote. | Logarithmic: Vertical asymptote.
Compare the domains and ranges of $f(x) = b^x$ and $g(x) = \log_b(x)$.
Exponential: Domain is $(-\infty, \infty)$, Range is $(0, \infty)$. | Logarithmic: Domain is $(0, \infty)$, Range is $(-\infty, \infty)$.
Compare the growth rates of exponential and logarithmic functions.
Exponential: Grows rapidly as $x$ increases. | Logarithmic: Grows slowly as $x$ increases.
Compare the behavior of $b^x$ and $\log_b(x)$ as $x$ approaches infinity.
Exponential: Approaches infinity. | Logarithmic: Approaches infinity, but much slower.
What does the graph of $y = b^x$ (where $b > 1$) look like?
An increasing curve that passes through (0, 1), approaching the x-axis as x goes to negative infinity.
What does the graph of $y = \log_b(x)$ (where $b > 1$) look like?
An increasing curve that passes through (1, 0), approaching the y-axis as x goes to 0.
How can you identify the base $b$ from the graph of $y = b^x$?
Find the point where $x = 1$. The $y$-coordinate of that point is the value of $b$.
How can you identify the base $b$ from the graph of $y = \log_b(x)$?
Find the point where $y = 1$. The $x$-coordinate of that point is the value of $b$.
What does the intersection of $y = b^x$ and $y = \log_b(x)$ with $y=x$ represent?
It shows the points where the function and its inverse have the same x and y values.
What is the general form of an exponential function?
$f(x) = ab^x$
What is the general form of a logarithmic function?
$f(x) = a \log_b(x)$
If $(s, t)$ is on $g(x) = b^x$, what point is on its inverse?
$(t, s)$ is on $f(x) = \log_b(x)$
Express $t = b^s$ in logarithmic form.
$s = \log_b(t)$
Express $s = \log_b(t)$ in exponential form.
$t = b^s$