Logarithmic functions

Summary

• For a given \(x∈(0,∞)\) we define a function \(y=log x\) called the natural (logarithm) as follows:
• Find the unique value \(y\) such that \(e^y=x\) . This is the value for \(log x\)
• The natural domain for \(log x\) is \((0,∞)\) , the range is \(R\)
• \(log⁡x\) is s strictly increasing function
• \(log x<0\) for \(0<x<1\)
• \(log 1=0\)
• \(log x>0\)
• Alternative notation: \(y=ln x\)
• Logarithm rules:
• \(e^{log x}=x\) for all \(x>0\)
• \(log e^x=x\) for all \(x\)
• \(log xy=log x+log y\) for \(x>0,y>0\)
• \(log x/y=log x-log y\) for \(x>0,y>0\)
• \(log x^y=ylog x\) for \(x>0\)