Transformations of Logarithmic Functions

Transformations apply to logarithmic functions the same way as they do to other functions. Recall what we have learnt in previous chapters on how transformations effect a function and its graph.

• f(x) → f(x) + c
• f(x) → f(x - d)
• f(x) → af(x)
• f(x) → f(kx)

The same techniques is applied to logarithmic functions. An equation of a transformed logarithmic function has a form of f(x) = a log_B [k (x+d)] + c where a, k, d and c are coefficients while B is the base of the log.

We must first apply the horizontal/vertical stretches or compressions first if any.

• y = log (kx) : Horizontal compression/stretch by factor of |1/k|
• y = a log x   : Vertical stretch/compression by a factor of |a|

The next step is to apply any reflections. If a <0, reflect in the x-axis. If k<0, reflect in the y-axis.

Then apply the horizontal or vertical translations to the function.

• y = f(x − d)  :  shift to the right by d units if d>0; translate left by d units if d<0
• y = log x + c : translate up by c units if c>0; shifts down by c units if c<0

Try answering these multiple choice questions on determining the shape of a graph when the logarithmic function is transformed. Hopefully it helps!