Math Help

Logarithmic and Exponential Functions

In a Nut Shell: Logarithmic functions and exponential functions provide the inverse of each

other. Let y = a^x where a^x is the exponential function with “base” a. Then the logarithmic

function with “base” a is log_a(y) = x.

The three most common types of logarithms are the binary logarithm with base “2”, the natural

logarithm with base “e”, and the common logarithm with base “10”. Logarithmic/exponential

functions occur in engineering applications such as in growth and decay of current in an

electronic circuit, in the decay of radioactive materials, and others.

The top figure below shows the graph of the logarithmic function, y = ln(x). The three figures

below it show the graphs of the exponential function y = a^x for differing values of a.

Properties of logarithms you need to know: a is “any” base

log_a(xy) = log_a (x) + log_a (y) , log_a (x/y) = log_a (x) – log_a (y)

log_a (x^r) = r log_a (x) , log_e(x) = ln(x) , log_a(a^x) = x , ln(e^x) = x , ln(e^–x) = – x

log_a (x) = log_K(x) / log_K(a) , log_a (x) = ln(x) / ln(a)

Properties of exponentials you need to know: y = a^x

a ^x+y = a^x a^y , a ^x–y = a^x / a^y , ( a ^x) ^y = a^x ^y , ( ab ) ^x = a^x b^x