Logarithm Summary: Properties, Formulas, Laws
Logarithms and logarithmic functions have been thoroughly covered in:
- Introduction to Logarithms
- Properties of Logarithms
- Change of Base Formula for Logarithms
- Introduction to Logarithmic Functions
- Logarithmic Functions: Review and Additional Properties
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=xRead ‘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 3−1=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 b≠1.
Allowable Bases
For Logarithms:
b>0, b≠1
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:
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
Function View Of Logarithms
Laws of Logarithms
Let b>0, b≠1, x>0, and y>0.
logbxy=logbx+logby
The log of a product is the sum of the logs.
logbxy=logbx−logby
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=logaxlogabIn 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, b≠1, x>0 and y>0:
x=y ⟺ logbx=logbyInverse 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≠1, and all real numbers x:
logbbx=xFor b>0, b≠1, and x>0:
blogbx=xSpecial Points
For b>0 and b≠1:
-
logbb=1 (since b1=b)
Equivalently, the point (b,1) lies on the graph of y=logbx.
-
logb1=0 (since b0=1)
Equivalently, the point (1,0) lies on the graph of y=logbx.
Graphs of Logarithmic Functions
Properties shared by all logarithmic graphs, y=logbx:
- 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=logbx is y=bx
- The graph of the inverse is shown in red on the graphs below: the inverse is the reflection of the logarithmic graph about the line y=x.
For b>1:
- y=logbx is an increasing function: x<y ⟺ logbx<logby
- Right-hand end behavior: as x→∞, y→∞
- x=0 is a vertical asymptote: as x→0+, y→−∞
b>1:
Blue curve: y=logbx
Red curve: inverse y=bx
Dashed line: y=x
For 0<b<1:
- y=logbx is a decreasing function: x<y ⟺ logbx>logby
- Right-hand end behavior: as x→∞, y→−∞
- x=0 is a vertical asymptote: as x→0+, y→∞
0<b<1:
Blue curve: y=logbx
Red curve: inverse y=bx
Dashed line: y=x
Concept Practice
- Choose a specific problem type, or click ‘New problem’ for a random question.
- Think about your answer.
- Click ‘Check your answer’ to check!