LOGARITHM SUMMARY: PROPERTIES, FORMULAS, LAWS

Logarithms and logarithmic functions have been thoroughly covered in:

This section provides an in-a-nutshell, at-a-glance summary of key results.

What is $\,\log_b x\,$? Logarithms are exponents!

The number  ‘$\,\log_b x\,$’  is the power (the exponent) that $\,b\,$ must be raised to,
in order to get $\,x\,$.
$$\log_b x = y\ \ \iff\ \ b^y = x$$ Read  ‘$\,\log_b x\,$’  as  ‘log base $\,b\,$ of $\,x\,$’ .

The equation  ‘$\,\log_b x = y\,$’  is called the logarithmic form of the equation.

The equation  ‘$\,b^y = x\,$’  is called the exponential form of the equation.
$\log_2 8 = 3\,$ since $\,2^3 = 8\,$

$\log_7 1 = 0\,$ since $\,7^0 = 1\,$

$\log_3 \frac 13 = -1\,$
since $\,3^{-1} = \frac 13\,$
Allowable Bases
for Logs
In the expression  ‘$\,\log_b x\,$’ , the number $\,b\,$ is called the base of the logarithm.

The number $\,b\,$ must be positive and not equal to $\,1\,$:   $\,b > 0\,$ and $\,b\ne 1\,$


allowable bases
for logarithms:
$b > 0\,$, $\,b\ne 1$
Two Special Logarithms The function  ‘$\,\log_{10}\,$’  (log base $\,10\,$) is called the common logarithm.
It is often abbreviated as just  ‘$\,\log\,$’  (with no indicated base).

The function  ‘$\,\log_{\text{e}}\,$’  (log base $\,\text{e}\,$) is called the natural logarithm.
It is often abbreviated as  ‘$\,\ln$’  (with no indicated base).

Caution: Some disciplines use  ‘$\,\log\,$’  to mean the natural logarithm.
Always check notation.
$\log x$
the common log of $\,x$

$\ln x$
the natural log of $\,x$
Allowable Inputs
for Logs
In the expression  ‘$\,\log_b x\,$’ , the number $\,x\,$ (the input) must be positive: $\,x > 0\,$

allowable inputs
for logarithms:
$x > 0$
Function View
of Logs
The number ‘$\log_b x\,$’ is the output from the function ‘$\,\log_b\,$’ when the input is ‘$\,x\,$’.

The domain of the function $\,\log_b\,$ is the set of all positive real numbers:   $\,\text{dom}(\log_b) = (0,\infty)$

The range of the function $\,\log_b\,$ is the set of all real numbers:
$\,\text{ran}(\log_b) = \Bbb R$


function view
of logarithms
Laws of Logarithms Let $\,b\gt 0\,$, $\,b\ne 1\,$, $\,x\gt 0\,$, and $\,y\gt 0\,$.

$\log_b\,xy = \log_b x + \log_b y$
The log of a product is the sum of the logs.

$\displaystyle \log_b\frac{x}{y} = \log_b x - \log_b y$
The log of a quotient is the difference of the logs.

For this final property, $\,y\,$ can be any real number:
$\log_b\,x^y = y\,\log_b x$
You can bring exponents down.

See
Properties of Logarithms
for a typical proof
of these laws.
Change of Base
Formula
for Logarithms
Let $\,a\,$ and $\,b\,$ be positive numbers that are not equal to $\,1\,$,
and let $\,x\gt 0\,$. Then, $$ \log_b\,x = \frac{\log_a\,x}{\log_a\,b} $$ In words:
You can change from any base $\,b\,$ to any base $\,a\,$;
the ‘adjustment’ is that you must divide by
the log to the new base ($\,a\,$)
of the old base ($\,b\,$).
See
Change of Base Formula for Logarithms
for a derivation of this formula.

The equation $$\,\log_b x = \left(\frac 1{\log_a b}\right)(\log_a x)\,$$ shows that any log curve
is just a vertical scaling
of any other log curve!
Logarithm functions
are one-to-one
Since logarithms are functions:
$x = y \ \ \Rightarrow\ \ \log_b x = \log_b y$
When inputs are the same, outputs are the same.

Since logarithms are one-to-one:
$\log_b x = \log_b y\ \ \Rightarrow\ \ x = y$
When outputs are the same, inputs are the same.

Thus, for all $\,b > 0\,$, $\,b\ne 1\,$, $\,x > 0\,$ and $\,y > 0\,$: $$x = y\ \ \iff\ \ \log_b x = \log_b y$$
 
Inverse Properties Logarithmic functions are one-to-one, hence have inverses.

The inverse of the logarithmic function with base $\,b\,$
is the exponential function with base $\,b\,$.

For $\,b > 0\,$, $\,b\ne 1\,$, and all real numbers $\,x\,$: $$\log_b b^x = x$$ For $\,b > 0\,$, $\,b \ne 1\,$, and $\,x > 0\,$: $$b^{\log_b x} = x$$
 
Special Points For $\,b > 0\,$ and $\,b\ne 1\,$:
  • $\log_b b = 1\,$   (since $\,b^1 = b\,$)
    Equivalently, the point $\,(b,1)\,$ lies on the graph of $\,y = \log_b x\,$.
  • $\log_b 1 = 0\,$   (since $\,b^0 = 1\,$)
    Equivalently, the point $\,(1,0)\,$ lies on the graph of $\,y = \log_b x\,$.
 


Graphs of Logarithmic Functions



Properties shared by all logarithmic graphs, $\,y = \log_b x\,$:
  • vertical asymptote: $\,x = 0$
  • pass both horizontal and vertical line tests
  • contain the point $\,(1,0)\,$
  • contain the point $\,(b,1)\,$
  • domain is the set of all positive real numbers
  • range is the set of all real numbers
  • the inverse of  $\,y = \log_b x\,$  is  $\,y = b^x\,$
  • the graph of the inverse
    is shown in red on the graphs at right:
    the inverse is the reflection of the logarithmic graph
    about the line $\,y = x\,$
For $\,b > 1\,$:
  • $y = \log_b x\,$ is an increasing function: $$x < y\ \ \iff\ \ \log_b x < \log_b y$$
  • right-hand end behavior:
    as $\,x\rightarrow\infty\,$, $\,y\rightarrow\infty$
  • $\,x = 0\,$ is a vertical asymptote:
    as $\,x\rightarrow 0^{+}\,$, $\,y\rightarrow -\infty\,$

$\,b > 1\,$:
blue curve: $\,y = \log_b x\,$
red curve: inverse $\,y = b^x\,$
dashed line: $\,y = x\,$
For $\,0 < b < 1\,$:
  • $y = \log_b x\,$ is a decreasing function: $$x < y\ \ \iff\ \ \log_b x > \log_b y$$
  • right-hand end behavior:
    as $\,x\rightarrow\infty\,$, $\,y\rightarrow -\infty$
  • $\,x = 0\,$ is a vertical asymptote:
    as $\,x\rightarrow 0^{+}\,$, $\,y\rightarrow \infty\,$

$\,0 < b < 1\,$:
blue curve: $\,y = \log_b x\,$
red curve: inverse $\,y = b^x\,$
dashed line: $\,y = x\,$
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Exponential Growth and Decay—Introduction
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
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