Transformations of logarithmic functions

Exponential functions and logarithms: Logarithms

Transformations of logarithmic functions

Transformation of a logarithmic function By horizontal and vertical displacement, and by vertical scaling of the graph of the logarithmic function \(f(x)=\log_\gamma(x)\) with base \(\gamma\) the graph of the function \[g(x)=a\cdot \log_\gamma(x+c) + d\quad\text{with parameters }a, c, d\text.\] can be created. In this way an arbitrary extended logarithmic relationship can be created.

We speak of an extended logarithmic relationship when \(d\neq0\). For \(d=0\) we have a 'normal' logarithmic relationship of the form \(y=a\cdot \log_\gamma(x+c)\) for certain constants \(a\), \(\gamma\) and \(c\). Using the calculation rules for logarithms you can even limit the number of constants needed to two: \(y=\log_h(x+c)\) with base \(h\) chosen such that \(h^a=\gamma\) and for a constant \(c\).

To illustrate transformations of logarithmic function, we considers at the horizontal and vertical shift as well as the vertical scaling of the graph of the logarithmic function \(f(x)=\log_\gamma(x)\) for a certain base \(\gamma\), which you can set via a slider.

\(\phantom{x}\)

Vertical translation

We translate the graph of \(f(x)=\log_\gamma(x)\) vertically over a distance \(\blue{d}\).

The new function becomes \[g(x)=\log_\gamma(x)+\blue{d}\] The vertical asymptote \(x=0\), the domain \((0,\infty)\) and the range \((-\infty,\infty)\) of \(g\) stay the same to those of \(f\).

GeoGebra

\(\phantom{x}\)

Horizontal translation

We translate the graph of \(f(x)=\log_\gamma(x)\) horizontally to the left with \(\blue{c}\).

The new function will be \[g(x)=\log_\gamma(x+\blue{c})\] The vertical asymptote of \(g\) is the line \(x=-\blue{c}\). The domain of \(g\) is equal to \((-\blue{c},\infty)\) The range of \(g\) is equal to the range of the logarithmic function \(f\), namely \((-\infty,\infty)\).

GeoGebra

\(\phantom{x}\)

Vertical scaling

We multiply the graph of \(f(x)=\log_\gamma(x)\) with respect to the \(x\)-axis by \(\blue{a}\).

The new function becomes \[g(x)=\blue{a}\cdot \log_\gamma(x)\] The vertical asymptote \(x=0\), the domain \((0,\infty)\) and the range \((-\infty,\infty)\) of \(g\) stay the same to those of \(f\). In fact, vertical scaling does nothing with the function \(f(x)=\log_\gamma(x)\) other than a logarithmic function with a different base: choose a number \(h\) such that \(h^\blue{a}=\gamma\), or \(\log_h(\gamma)=\blue{a}\) and then \(\blue{a}\cdot \log_\gamma(x)=\log_h(x)\) according to the calculation rules for logarithms.

GeoGebra

Horizontal scaling Horizontal scaling, i.e., multiplication with respect to the \(y\)-axis with scalar \(\frac{1}{b}\), is not included in the overview of transformations because, according to the calculation rules of logarithms, in the case \(b>0\) it is actually a vertical translation. If \(y=\log_\gamma(b\cdot x)\) then \(y=\log_\gamma(x)+ \log_\gamma(b)\) applies. If \(b<0\), then the transform is a vertical translation of the mirrored function \(\log_\gamma(-x)\). In the latter case, the domain of the function consists of all negative real numbers.

Interactive example You can combine the above transformations to construct any extended logarithmic relationship. In the interactive example below you can explore via the sliders the effect of the parameters \(\color{red}{a}\), \(\color{blue}{c}\), \(\color{magenta}{d}\), and \(\color{green}{\gamma}\) that lead to the graph of the function \[g(x)=\color{red}{a}\cdot \log_{\color{green}{\gamma}}(x+\color{blue}{c})+\color{magenta}{d}\] via the sliders.

GeoGebra