For a>0, a 1, the functions ax and log_a x are inverses. Their graphs reflect in y=x. The special base e gives ex and ln x. **Function** | **Domain; range** | **Key features** ax | R; (0,infinity) | (0,1); asymptote y=0 log_a x | (0,infinity); R | (1,0); asymptote x=0 Both increase if a>1 and decrease if 0<a<1. Also ax=exln a and log_a(ax)=x for every real x. Example y=ln(x-2) has domain x>2, range R, asymptote x=2, intercept (3,0), and no y-intercept.

For \(a>0\), \(a\ne1\), the functions \(a^x\) and \(\log_a x\) are inverses. Their graphs reflect in \(y=x\). The special base \(e\) gives \(e^x\) and \(\ln x\).

Both increase if \(a>1\) and decrease if \(0<a<1\). Also \(a^x=e^{x\ln a}\) and \(\log_a(a^x)=x\) for every real \(x\).

\(y=\ln(x-2)\) has domain \(x>2\), range \(\mathbb R\), asymptote \(x=2\), intercept \((3,0)\), and no \(y\)-intercept.

Exponential and Logarithmic Functions

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Exponential and logarithmic graphs

Exponential and logarithmic graphs