# Logarithms

amath
2^x=8 has solution x=3. What about 2^x=7 ?

Logarithms come in handy when exponential equations do not have rational solutions.

a^x=b if and only if log_a b=x for a,b >0 and a \ne 1

Solve the following examples using the definition of logarithms:

1. Find log_5 1

step endamath 1 : amath log_5 1=x

step endamath 2 : amath 5^x=1

step endamath 3 : amath x=0

2. Find log_6 6 sqrt(6)

step endamath 1 : amath log_6 6 sqrt(6)=x

step endamath 2 : amath 6^x=6 sqrt(6)=6^(3/2)

step endamath 3 : amath x=3/2

The natural logarithm (ln) is the logarithm with e as its base.

ln (1/e)=-1

The common logarithm is the logarithm with base 10. We do not write the subscript when the base is 10.

log 0.01=-2

The graph of y=2^x is an increasing function with domain = all reals and range= positive reals.
The graph of y=(1/2)^x is a decreasing function with domain = all reals and range= positive reals.

## How to find the inverse of f(x)=b^x ?

y=b^x

x=b^y

Isolating y, we have y=f^(-1)(x)=log_b x.

Thus, the inverse of f(x)=b^x is f^(-1)(x)=log_b x.

Domain= positive reals

Range=all reals

Download the MATLAB GUI files, logarithmicfunctions.m and logarithmicfunctions.fig to graph y=b^x and its inverse.

Here are some questions for you to test your knowledge on the basic definition of logarithms and on logarithmic functions.

Click here for laws on logarithms.
endamath