A core exponential function has the form $\exp_b(x) = b^x$ where $b > 0$, $b \ne 1$.

For example in ELF.0: $f(x) = \exp_2(x) = 2^x$.

A core logarithmic function has the form $\log_b(y)$ where $b > 0$, $b \ne 1$, and $ y \in (0,\infty)$.
For example in ELF.0 : $g(x) = \log_2(y)$.

We explore how mapping diagrams for core exponential and logarithmic functions help us visualize and understand further the meaning of the shape of the graphs of these functions .
Example ELF.CELF.1 : $b=3; f(x) = \exp_3(x) = 3^x$ and $g(x)=\log_3(x) $.
Example ELF.CELF.2 : $b= \frac 1 2 ;  f(x) =\exp_{\frac 1 2}(x) =  (\frac 1 2) ^x$ and $g(x)= \log_ {\frac 1 2} (x)$.
Example ELF.CELF.3 : $b=10;   f(x) =\exp_{10}(x) = 10^x$ and $g(x)=\log_{10}(x) $.
Example ELF.CELF.4 : $b=e;   f(x) =\exp_e(x)=e^x$ and $g(x)=\log_e(x)=ln(x) $.
Example ELF.DCELF.0 : Dynamic Visualization of Core Exponential and Logarithmic Functions: Graphs and Mapping Diagrams