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