Find the Range of \(f(x)=\log_a x\)
Question
Find the range of the function
\[ f(x)=\log_a x \]where
\[ a>0,\quad a\ne1 \]Solution
Given
\[ f(x)=\log_a x \]where
\[ a>0,\quad a\ne1 \]Domain of the Function
For logarithmic functions,
\[ x>0 \]Finding the Range
Let
\[ y=\log_a x \]Converting into exponential form,
\[ x=a^y \]Since \(a>0\) and \(a\ne1\), the expression \(a^y\) is defined for every real value of \(y\).
Therefore, corresponding to every real number \(y\), there exists a positive real number \(x\).
Hence \(y\) can take all real values.