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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.

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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 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 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.

$t = b^s$