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  2. AP Pre Calculus
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What does the graph of y=bxy = b^xy=bx (where b>1b > 1b>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=bxy = b^xy=bx (where b>1b > 1b>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)y = \log_b(x)y=logb​(x) (where b>1b > 1b>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 bbb from the graph of y=bxy = b^xy=bx?

Find the point where x=1x = 1x=1. The yyy-coordinate of that point is the value of bbb.

How can you identify the base bbb from the graph of y=log⁡b(x)y = \log_b(x)y=logb​(x)?

Find the point where y=1y = 1y=1. The xxx-coordinate of that point is the value of bbb.

What does the intersection of y=bxy = b^xy=bx and y=log⁡b(x)y = \log_b(x)y=logb​(x) with y=xy=xy=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)=abxf(x) = ab^xf(x)=abx

What is the general form of a logarithmic function?

f(x)=alog⁡b(x)f(x) = a \log_b(x)f(x)=alogb​(x)

If (s,t)(s, t)(s,t) is on g(x)=bxg(x) = b^xg(x)=bx, what point is on its inverse?

(t,s)(t, s)(t,s) is on f(x)=log⁡b(x)f(x) = \log_b(x)f(x)=logb​(x)

Express t=bst = b^st=bs in logarithmic form.

s=log⁡b(t)s = \log_b(t)s=logb​(t)

Express s=log⁡b(t)s = \log_b(t)s=logb​(t) in exponential form.

t=bst = b^st=bs

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)=bxf(x) = b^xf(x)=bx and g(x)=log⁡b(x)g(x) = \log_b(x)g(x)=logb​(x).

Exponential: Domain is (−∞,∞)(-\infty, \infty)(−∞,∞), Range is (0,∞)(0, \infty)(0,∞). | Logarithmic: Domain is (0,∞)(0, \infty)(0,∞), Range is (−∞,∞)(-\infty, \infty)(−∞,∞).

Compare the growth rates of exponential and logarithmic functions.

Exponential: Grows rapidly as xxx increases. | Logarithmic: Grows slowly as xxx increases.

Compare the behavior of bxb^xbx and log⁡b(x)\log_b(x)logb​(x) as xxx approaches infinity.

Exponential: Approaches infinity. | Logarithmic: Approaches infinity, but much slower.