AP PreCalculus Practice Quiz: Logarithmic Function Manipulation

Correct Answer: A

According to the product property for logarithms, log_{b}(xy) = log_{b}x + log_{b}y. Applying this property in reverse, log_{b}(5) + log_{b}(x) is equivalent to log_{b}(5x).

Correct Answer: B

The power property for logarithms states that n log_{b}x = log_{b}x^{n}. In this case, n=4, so 4 log_{b}(x) is equivalent to log_{b}(x^{4}).

Correct Answer: A

The provided content states that the product property, log_{b}(xy) = log_{b}x + log_{b}y, implies that every horizontal dilation of a logarithmic function is equivalent to a vertical translation. For example, log_{b}(kx) = log_{b}(k) + log_{b}(x), where log_{b}(k) is a constant vertical shift.

Correct Answer: B

The content explicitly states that the power property, log_{b}x^{n} = n log_{b}x, implies that raising the input of a logarithmic function to a power results in a vertical dilation. The original function log_{b}x is multiplied by a factor of n.

Correct Answer: C

The change of base property is given as log_{b}x = log_{a}x / log_{a}b. Since 1/log_{a}b is a constant, this means that any logarithmic function log_{b}x is just a constant multiple of any other logarithmic function log_{a}x. This relationship is a vertical dilation.

Correct Answer: B

The change of base property is log_{b}x = log_{a}x / log_{a}b. To change log_{5}(x) to the natural base e, we set b=5 and a=e. This gives log_{e}x / log_{e}5, which is equivalent to ln(x) / ln(5).

Correct Answer: C

The provided content defines the natural logarithm, ln x, as a logarithmic function with the natural base e. Therefore, ln x = log_{e}x.

Correct Answer: C

This requires two properties. First, use the power property to rewrite 2 log_{b}(x) as log_{b}(x^{2}). Then, use the product property to combine log_{b}(x^{2}) + log_{b}(y) into log_{b}(x^{2}y).

Correct Answer: A

The product property states that a horizontal dilation is equivalent to a vertical translation. By rewriting log_{3}(9x) as 2 + log_{3}(x), we see that the graph of y = log_{3}(x) has been shifted vertically by +2 units.

Correct Answer: D

Using the power property, g(x) = ln(x^{5}) = 5 ln(x), which is a vertical dilation of f(x) by a factor of 5. Using the change of base property, h(x) = log_{7}(x) = ln(x) / ln(7), which is a vertical dilation of f(x) by a factor of 1/ln(7). Therefore, both are vertical dilations of f(x).

Correct Answer: C

First, apply the product property: ln(x^{a}y^{b}) = ln(x^{a}) + ln(y^{b}). Then, apply the power property to both terms: ln(x^{a}) + ln(y^{b}) = a ln x + b ln y.