Explain the inverse relationship between exponential and logarithmic functions.

Logarithmic functions 'undo' exponential functions. The input of one is the output of the other.

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Explain the inverse relationship between exponential and logarithmic functions.

Logarithmic functions 'undo' exponential functions. The input of one is the output of the other.

How does exponential growth change with input values?

Output values change multiplicatively as input values change additively.

How does logarithmic growth change with input values?

Output values change additively as input values change multiplicatively.

Describe the reflection of an exponential function over $y=x$ .

It results in the graph of its inverse, a logarithmic function.

What are the key characteristics of exponential functions?

Rapid increase as $x$ increases (if $b > 1$ ), vertical asymptote at $x=0$ , no horizontal asymptote.

What are the key characteristics of logarithmic functions?

Slow increase as $x$ increases (if $b > 1$ ), horizontal asymptote at $y=0$ , no vertical asymptote.

How are the domains and ranges of exponential and logarithmic functions related?

The domain of an exponential function is the range of its inverse logarithmic function, and vice versa.

What is the significance of the line $y = x$ when graphing inverse functions?

It acts as a 'mirror' across which the graphs of the function and its inverse are reflected.

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.