Applications of logarithms and exponential functions appear almost immediately after these functions are introduced as early as beginning algebra,

The most frequent application of logarithms historically and before the advent of electronic calculators was for scientific calculations and understanding the operation of slide rules.

After calculators and computers made computation much easier, the exponential and logarithm functions continue to have great importance as functions for modeling growth and decay in many contexts,

Financial models entailing continuous compounding use the natural exponential model; population models use simple exponential models as well as more subtle logistic models; and the study of radioactive decay and newton's law of cooling both lead to exponential functions that model decreasing functions.

Logarithmic functions arise frequently as well, to solve equations related to exponential functions and in the derivation of exponential functions as models from differential equations.

Visualizing the information used in these applications is sometimes connected to Cartesian graphs or often ignored with only algebraic treatments presented.

In this section examples are given to illustrate how mapping diagrams can contribute a visual approach to these applications.

Applications of these functions have three canonical problems:

- Given the modeling function, $f$ that depends on time, $t$,
__find the value__**at a particular time, $t*$,**that is find $f(t*)$. [Evaluate the function]

- Given the modeling function, $f$ that depends on time, $t$, and
**a value of that function, $v$**,__find any time__, $t_v$, where $f(t_v) = v$. [Solve the equation] - Given an exponential form of a modeling function with undetermined
parameters, $f$ that depends on time, $t$, and some value(s) at
particular time (s), such as $f(t_1) = v_1$ and $f(t_2)= v_2$, find the
parameters of $f$ useful in solving the previous problems.