What does the graph of $y = \log_b(x)$ with b > 1 tell us?

The function is increasing, and the graph is concave down.

What does the graph of $y = \log_b(x)$ with 0 < b < 1 tell us?

The function is decreasing, and the graph is concave down.

How does a vertical asymptote appear on the graph of a logarithmic function?

As a vertical line that the graph approaches but never crosses, indicating a domain restriction.

How does a horizontal shift affect the graph of a logarithmic function?

It moves the entire graph left or right, changing the position of the vertical asymptote.

What does a reflection across the x-axis do to the graph of a logarithmic function?

It inverts the function, changing increasing functions to decreasing and vice versa.

How can you identify the base of a logarithmic function from its graph?

Look for a point (x, y) on the graph and solve for b in the equation $b^y = x$ .

What does the steepness of a logarithmic graph indicate?

It indicates the rate of change of the function. Steeper graphs have a faster rate of change near the asymptote.

How does the graph of $y = \log_b(x)$ relate to the graph of $y = b^x$ ?

They are reflections of each other across the line y = x.

What does a vertical stretch of a logarithmic function look like on its graph?

The graph appears to be stretched vertically away from the x-axis.

How can you determine the domain of a transformed logarithmic function from its graph?

Identify the vertical asymptote; the domain is all x-values greater than (or less than, depending on reflection) the asymptote's x-value.

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 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: $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$

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