AP®︎ Precalculus: Topic 2.12 Practice Test

Essential Knowledge: 2.12.A.1

The product property for logarithms states that $\log_{b}(xy) = \log_{b} x + \log_{b} y$. Graphically, this property implies that every horizontal dilation of a logarithmic function, $f(x) = \log_{b}(kx)$, is equivalent to a vertical translation, $f(x) = \log_{b}(kx) = \log_{b} k + \log_{b} x = a + \log_{b} x$, where $a = \log_{b} k$.

Essential Knowledge: 2.12.A.2

The power property for logarithms states that $\log_{b} x^{n} = n\log_{b} x$. Graphically, this property implies that raising the input of a logarithmic function to a power, $f(x) = \log_{b} x^{k}$, results in a vertical dilation, $f(x) = \log_{b} x^{k} = k\log_{b} x$.

Essential Knowledge: 2.12.A.3

The change of base property for logarithms states that $\log_{b} x = \frac{\log_{a} x}{\log_{a} b}$, where $a > 0$ and $a \ne 1$. This implies that all logarithmic functions are vertical dilations of each other.

Essential Knowledge: 2.12.A.4

The function $f(x) = \ln x$ is a logarithmic function with the natural base $e$; that is, $\ln x = \log_{e} x$.