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




2. Find log_6 6 sqrt(6)




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