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audio read-through 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.

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What is logbx?

Logarithms are exponents! The number  logbx  is the power (the exponent) that b must be raised to, in order to get x.

logbx=y    by=x

Read  logbx  as  ‘log base b of x’.

The equation  logbx=y  is called the logarithmic form of the equation.

The equation  by=x  is called the exponential form of the equation.

Examples:

log28=3  since  23=8

log71=0  since  70=1

log313=1  since  31=13

Allowable Bases For Logs

In the expression  logbx’ , the number b is called the base of the logarithm. The number b must be positive and not equal to 1:  b>0 and b1.

allowable bases for logarithms

Allowable Bases For Logarithms:
b>0, b1

Two Special Logarithms

The function  log10   (log base 10) is called the common logarithm. It is often abbreviated as just  log  (with no indicated base).

The function  loge   (log base 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.

Summary:

logx:  the common log of x

lnx:  the natural log of x

Allowable Inputs For Logs

In the expression  logbx’, the number x (the input) must be positive: x>0.

allowable inputs for logarithms

Allowable Inputs For Logarithms:
x>0

Function View of Logs

The number  logbx  is the output from the function  logb  when the input is  x’.

The domain of the function logb is the set of all positive real numbers:   dom(logb)=(0,)

The range of the function logb is the set of all real numbers:   ran(logb)=R

the function log_b

Function View Of Logarithms

Laws of Logarithms

Let b>0, b1, x>0, and y>0.

logbxy=logbx+logby
The log of a product is the sum of the logs.

logbxy=logbxlogby
The log of a quotient is the difference of the logs.

For this final property, y can be any real number:

logbxy=ylogbx
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>0. Then:

logbx=logaxlogab

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

logbx=(1logab)(logax)

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    logbx=logby
When inputs are the same, outputs are the same.

Since logarithms are one-to-one:

logbx=logby    x=y
When outputs are the same, inputs are the same.

Thus, for all b>0, b1, x>0 and y>0:

x=y    logbx=logby

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, b1, and all real numbers x:

logbbx=x

For b>0, b1, and x>0:

blogbx=x

Special Points

For b>0 and b1:

Graphs of Logarithmic Functions

Properties shared by all logarithmic graphs, y=logbx:

For b>1:

log function and inverse for base greater than 1

b>1:
Blue curve: y=logbx
Red curve: inverse y=bx
Dashed line: y=x

For 0<b<1:

log function and inverse for base between 0 and 1

0<b<1:
Blue curve: y=logbx
Red curve: inverse y=bx
Dashed line: y=x

Master the ideas from this section by practicing below:

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When you're done practicing, move on to:

Exponential Growth and Decay—Introduction
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Concept Practice

  1. Choose a specific problem type, or click ‘New problem’ for a random question.
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  3. Click ‘Check your answer’ to check!
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