A logarithmic function has the form $y=k\log_a x$, where $a>0$, $a\ne1$ and $k\ne0$. With $a=e$ and $k=1$, this is the natural logarithmic function $y=\ln x$. Note: The input $x$ is always positive ($x>0$); the output $y$ can take any real values. $y=k{{\log }_{a}}x$ $k=1$ $k=-1$ Notes $a>1$ It is a logarithmic graph. The curve cuts the $x$-axis at $\left( 1,0 \right)$. The $y$-axis is the vertical asymptote. For $k>0$, the graph increases gently as $x$ increases. For $k<0$, the graph decreases gently as $x$increases. $0<a<1$ It is a logarithmic graph. The curve cuts the $x$-axis at $\left( 1,0 \right)$. The $y$-axis is the vertical asymptote. For $k>0$, the graph decreases gently as $x$ increases. For $k<0$, the graph increases gently as $x$increases. Relationship among Graphs of Logarithmic Functions with Different Bases For logarithmic functions in the form of $y={{\log }_{a}}x$: For fixed $x>1$, increasing $a$ decreases $\log_a x$, within either base interval $0<a<1$ or $a>1$. For fixed $0<x<1$, increasing $a$ increases $\log_a x$, within either base interval. They all pass through the point $\left( 1,0 \right)$. They all have a vertical asymptote as the $y$-axis. Relationship between Graphs of Exponential and Logarithmic Functions with the Same Base For exponential and logarithmic functions in the forms of $y={{a}^{x}}$ and $y={{\log }_{a}}x$: Their graphs are symmetrical about the straight line $y=x$. The functions $y={{a}^{x}}$ and $y={{\log }_{a}}x$ are the inverse functions of each other.

Logarithmic Functions and Graphs

IP 3

A logarithmic function has the form $y=k\log_a x$, where $a>0$, $a\ne1$ and $k\ne0$. With $a=e$ and $k=1$, this is the natural logarithmic function $y=\ln x$.

Note: The input $x$ is always positive ($x>0$); the output $y$ can take any real values.

$y=k{{\log }_{a}}x$	$k=1$	$k=-1$	Notes
$a>1$			It is a logarithmic graph. The curve cuts the $x$-axis at $\left( 1,0 \right)$. The $y$-axis is the vertical asymptote. For $k>0$, the graph increases gently as $x$ increases. For $k<0$, the graph decreases gently as $x$increases.
$0<a<1$			It is a logarithmic graph. The curve cuts the $x$-axis at $\left( 1,0 \right)$. The $y$-axis is the vertical asymptote. For $k>0$, the graph decreases gently as $x$ increases. For $k<0$, the graph increases gently as $x$increases.

Relationship among Graphs of Logarithmic Functions with Different Bases
	For logarithmic functions in the form of \(y={{\log }_{a}}x\): For fixed \(x>1\), increasing \(a\) decreases \(\log_a x\), within either base interval \(0<a<1\) or \(a>1\). For fixed \(0<x<1\), increasing \(a\) increases \(\log_a x\), within either base interval. They all pass through the point \(\left( 1,0 \right)\). They all have a vertical asymptote as the \(y\)-axis.
Relationship between Graphs of Exponential and Logarithmic Functions with the Same Base
	For exponential and logarithmic functions in the forms of \(y={{a}^{x}}\) and \(y={{\log }_{a}}x\): Their graphs are symmetrical about the straight line \(y=x\). The functions \(y={{a}^{x}}\) and \(y={{\log }_{a}}x\) are the inverse functions of each other.

Relationship among Graphs of Logarithmic Functions with Different Bases

For logarithmic functions in the form of $y={{\log }_{a}}x$:

For fixed $x>1$, increasing $a$ decreases $\log_a x$, within either base interval $0<a<1$ or $a>1$.
For fixed $0<x<1$, increasing $a$ increases $\log_a x$, within either base interval.
They all pass through the point $\left( 1,0 \right)$.
They all have a vertical asymptote as the $y$-axis.

Relationship between Graphs of Exponential and Logarithmic Functions with the Same Base

For exponential and logarithmic functions in the forms of $y={{a}^{x}}$ and $y={{\log }_{a}}x$:

Their graphs are symmetrical about the straight line $y=x$.
The functions $y={{a}^{x}}$ and $y={{\log }_{a}}x$ are the inverse functions of each other.

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\(y=k{{\log }_{a}}x\)	\(k=1\)	\(k=-1\)	Notes
\(a>1\)			It is a logarithmic graph. The curve cuts the \(x\)-axis at \(\left( 1,0 \right)\). The \(y\)-axis is the vertical asymptote. For \(k>0\), the graph increases gently as \(x\) increases. For \(k<0\), the graph decreases gently as \(x\)increases.
\(0<a<1\)			It is a logarithmic graph. The curve cuts the \(x\)-axis at \(\left( 1,0 \right)\). The \(y\)-axis is the vertical asymptote. For \(k>0\), the graph decreases gently as \(x\) increases. For \(k<0\), the graph increases gently as \(x\)increases.