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The logarithmic expression log_b a is read
the logarithm of a with base b.Here, the base b is clearly written in the expression. There are two cases in which the base does not need to be written, which will be discussed in this lesson.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Explore
Mathematical Operations
The n^(th) root of a real number a expresses another real number that, when multiplied by itself n times, will result in a. Aside from the radical symbol, the notation is made up of the radicand a and the index n.
However, in the case of a square root, the index can be omitted. Therefore, if in the root there is no index written, it is implied that the index is 2.
sqrt(a)=sqrt(a)
Something similar happens with the exponent of a number. An exponent is a number or expression written above a number, indicating that it is being multiplied by itself. The notation is made up of a base — the number being multiplied — and an exponent, which gives the number of times the base appears in the multiplication. 
The resulting number is commonly called a power. In this example, the base 7 is multiplied by itself 4 times. 7^4 = 7 * 7 * 7 * 7_4 However, if the exponent of a number or expression is 1, it does not need to be written. a=a^1 In the case of logarithms, there are two bases that do not need to be written. Take a look at a calculator and try to identify them.
Discussion
Common Logarithm
A common logarithm is a logarithm of base 10. For example, log_(10) 1000 is called the common logarithm of 1000.
It is equal to 3 because 10^3 is 1000.
Since common logarithms are used so often, the base does not need to be written.
log m ≡ log_(10) m
In the identity above, m is a positive number. Recalling the definition of a logarithm, the common logarithm of m can be defined for positive values of m.
log m=n ⇔ 10^n=m
Common logarithms can be evaluated using a calculator. For example, to evaluate log 34, push LOG, enter 34, and then hit ENTER.
The common logarithm of 34 is about 1.53.
Pop Quiz
Evaluating Common Logarithms
Use a calculator to evaluate the common logarithms. Round the answer to two decimal places.
Example
Earthquakes and Logarithms
A well-known logarithmic scale that is used to express the amount of energy released by an earthquake is the Richter scale. 
On this scale, each integer number increase indicates an intensity ten times stronger than the previous whole number. For example, an earthquake of magnitude 3 on the Richter scale is 10 times stronger than an earthquake of magnitude 2. An earthquake of magnitude 4 is 10* 10 or 100 times stronger than an earthquake of magnitude 2. The intensity of an earthquake is determined by a seismograph.
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Hint
Substitute 7.9 and 7.1 for M_1 and M_2, respectively, in the given formula. Then, use the definition of a common logarithm.
Solution
To calculate how many times more intense the 1906 earthquake was compared to the one in 1989, the value of I_1I_2 needs to be found. To do so, 7.9 and 7.1 can be substituted for M_1 and M_2, respectively, in the given formula.
This equation can be rewritten by following the definition of a common logarithm.
log I_1/I_2=0.8 ⇔ I_1/I_2=10^(0.8)
Finally, use a calculator to evaluate 10^(0.8).
The 1906 earthquake was about 6.31 times more intense than the 1989 earthquake.
log I_1/I_2=M_1-M_2
SubstituteII
M_1= 7.9, M_2= 7.1
log I_1/I_2= 7.9- 7.1
SubTerm
Subtract term
log I_1/I_2=0.8
Example
Using the Properties of Common Logarithms
After studying the relationship between earthquakes and logarithms, Vincenzo became more interested in this fascinating math topic.
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Hint
Use the properties of logarithms and the definition of a common logarithm.
Solution
First, simplify the expressions on the left. They can then be paired with their corresponding expressions on the right.
Expression 1
To simplify this expression, the Quotient Property of Logarithms and the definition of a common logarithm will be used.log 10x - log x
log(m) - log(n)=log(m/n)
log 10x/x
CancelCommonFac
Cancel out common factors
log 10x/x
SimpQuot
Simplify quotient
log 10
log(10) = 1
1
Expression 2
To simplify this expression, the Power Property of Logarithms, the Product Property of Logarithms, and the definition of a common logarithm will be used.2log x+log 10x
m*log(a)=log(a^m)
log x^2+log 10x
log(mn)=log(m) + log(n)
log x^2+log 10+log x
log(10) = 1
log x^2+1+log x
CommutativePropAdd
Commutative Property of Addition
1+log x+log x^2
log(m) + log(n)=log(mn)
1+log (x* x^2)
MultBasePow
a*a^m=a^(1+m)
1+log x^3
Expression 3
To simplify the third expression, the Product Property of Logarithms, the Quotient Property of Logarithms, a logarithm identity, and the definition of a common logarithm will be used.log 0.1x
log(mn)=log(m) + log(n)
log 0.1+log x
NumberToFrac
a = 10* a/10
log 10* 0.1/10+log x
Multiply
Multiply
log 1/10+log x
log(m/n)=log(m) - log(n)
log 1-log 10+log x
log(1) = 0
0-log 10+log x
log(10) = 1
0-1+log x
SubTerm
Subtract term
- 1+log x
CommutativePropAdd
Commutative Property of Addition
log x-1
Expression 4
To simplify the second to last expression, the Quotient Property of Logarithms, the Power Property of Logarithms, and the definition of a common logarithm will be used.log 100/x+log x^2
log(m/n)=log(m) - log(n)
log 100- log x+log x^2
WritePow
Write as a power
log 10^2- log x+log x^2
log(a^m)= m*log(a)
2log 10- log x+2log x
log(10) = 1
2(1)- log x+2log x
IdPropMult
Identity Property of Multiplication
2- log x+2log x
AddTerms
Add terms
2+ log x
Expression 5
Finally, to simplify the last expression, the Product Property of Logarithms and the Power Property of Logarithms will be used.log x+log x^2-3log x
log(m) + log(n)=log(mn)
log (x* x^2)-3log x
MultBasePow
a*a^m=a^(1+m)
log x^3-3log x
m*log(a)=log(a^m)
log x^3-log x^3
SubTerm
Subtract term
0
Explore
Calculating the Value of an Expression for Bigger Values of n
Consider the following algebraic expression.
(1+1/n)^n
This expression can be evaluated for different values of n. Pay close attention to the value of the expression as n increases.
What conclusion can be made?
Discussion
The Natural Base
Logarithms and exponents are closely related. This part of the lesson will explore a particular type of exponential expression. Recall the formula for compound interest.
Here, r is the interest rate in decimal form. If the interest rate is 100 %, which is extremely profitable, the value of r equals 1.
A=P(1+ 1n)^(nt)
The number n is the amount of times the interest is compounded each year. This is how often the accrued interest is added to the balance. The more often the interest is compounded, the higher the profit will be each year. What happens as n increases? To examine this, the formula will be rewritten using the the Power of a Power Property.
A=P(1+ 1n)^(nt) ⇕ A=P((1+ 1n)^n)^t
In the revised formula, the expression in the outer parentheses is a constant that depends only on n. To analyze what happens as n increases, larger and larger values will be substituted into this expression.
As the values of n increase higher and higher, the expression seems to approach a value of about 2.718. In fact, as n goes to infinity, the value of the expression approaches a constant that is called the natural base, or e.
Euler's number e appears in several areas of mathematics and has multiple uses. For example, it is often used as the base of exponential and logarithmic functions.
Concept
The Number e
The number e — commonly called the natural base — is an irrational mathematical constant named by the mathematician Leonhard Euler.
e = 2.7182818284...
Discussion
The Natural Logarithm
A natural logarithm is a logarithm with base e.
Although it is correct to write log_e,
the natural logarithm is more commonly written as ln.
ln m = log_e m
This means that ln m equals the exponent to which e must be raised to equal m.
ln m=n ⇔ m = e^n
Natural logarithms can be evaluated using a calculator. For example, to evaluate ln 21, push LN, input 21, and then hit ENTER.
Pop Quiz
Evaluating Natural Logarithms
Use a calculator to evaluate the natural logarithms. Round the answer to two decimal places.
Example
Using the Properties of Natural Logarithms
Vincenzo already knows how to use the properties of logarithms when dealing with common logarithms. Now he wants to use the properties when natural logarithms are involved.
To do so, Vincenzo asked his math teacher for some extra practice. She asked him to identify the only expression that is not equivalent to the others. Help him do this!
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Hint
Simplify all the expressions using the properties of logarithms.
Solution
Simplify the expressions so that the expression that is not equivalent with the others can be easily identified.
Expression 1
To simplify this expression, the Product Property of Logarithms, the Quotient Property of Logarithms, and the Power Property of Logarithms will be used.ln x^2 +ln y/x+ln z
ln(a^b)= b*ln(a)
2ln x +ln y/x+ln z
ln(a/b)=ln(a) - ln(b)
2ln x +ln y-ln x+ln z
SubTerm
Subtract term
ln x +ln y+ln z
ln(a) + ln(b)=ln(ab)
ln xy+ln z
ln(a) + ln(b)=ln(ab)
ln xyz
Expression 2
To simplify the second expression, the Product Property of Logarithms, the Power Property of Logarithms, and the definition of a natural logarithm will be used.ln ex+ln y^2z-(ln y+1)
ln(ab)=ln(a) + ln(b)
ln e+ln x+ln y^2+ln z-(ln y+1)
ln(e) = 1
1+ln x+ln y^2+ln z-(ln y+1)
ln(a^b)= b*ln(a)
1+ln x+2ln y+ln z-(ln y+1)
Distr
Distribute - 1
1+ln x+2ln y+ln z-ln y-1
SubTerms
Subtract terms
ln x+ln y+ln z
ln(a) + ln(b)=ln(ab)
ln xy+ln z
ln(a) + ln(b)=ln(ab)
ln xyz
Expression 3
To simplify the third expression, the Quotient Property of Logarithms and the definition of a natural logarithm will be used.- ln e/xyz+1
ln(a/b)=ln(a) - ln(b)
- (ln e -ln xyz)+1
ln(e) = 1
- (1 -ln xyz)+1
Distr
Distribute - 1
- 1 +ln xyz+1
AddTerms
Add terms
ln xyz
Expression 4
To simplify the second to last expression, the Product Property of Logarithms and the Quotient Property of Logarithms will be used.ln xy+ln z-ln y
ln(a) + ln(b)=ln(ab)
ln xyz-ln y
ln(a) - ln(b)=ln(a/b)
ln xyz/y
CancelCommonFac
Cancel out common factors
ln xyz/y
SimpQuot
Simplify quotient
ln xz
Expression 5
The last expression will be simplified by using the Power Property of Logarithms and the Product Property of Logarithms.2ln sqrt(xy) +ln z^2 -ln z
SqrtToPowD
sqrt(a)=a^(12)
2ln (xy)^(12) +ln z^2 -ln z
ln(a^b)= b*ln(a)
2(1/2)ln xy +2ln z -ln z
DenomMultFracToNumber
2 * a/2= a
1ln xy +2ln z -ln z
IdPropMult
Identity Property of Multiplication
ln xy +2ln z -ln z
SubTerm
Subtract term
ln xy +ln z
ln(a) + ln(b)=ln(ab)
ln xyz
Discussion
Change of Base Formula
Most calculators only calculate common and natural logarithms. These are logarithms with base 10 or e. Luckily, there is a formula that allows any logarithm to be written in terms of common or natural logarithms.
Rule
Change of Base Formula
A logarithm of arbitrary base can be rewritten as the quotient of two logarithms with the same base by using the change of base formula.
log_c a= log_b a/log_b c
This rule is valid for positive values of a,b, and c, where b and c are different than 1.
Proof
Let x=log_c a. Therefore, by the definition of a logarithm, it is known that a=c^x.
x=log_c a ⇔ a=c^x
By the Reflexive Property of Equality, log_b a is equal to itself.
log_b a=log_b a
Substitute
a= c^x
log_b a=log_b c^x
log_b(a^m)= m* log_b(a)
log_b a=xlog_b c
Substitute
x= log_c a
log_b a= log_c alog_b c
DivEqn
.LHS /log_b c.=.RHS /log_b c.
log_b a/log_b c=log_c a
RearrangeEqn
Rearrange equation
log_c a=log_b a/log_b c ✓
Note that this formula is helpful to calculate any logarithm using a calculator, since the new base can be any positive number different than 1. This means that the new base can be 10 or e.
log_c a=log a/log c log_c a=ln a/ln cExample
Using the Change of Base Formula
Vincenzo has discovered that he can use the Change of Base Formula to calculate the value of any logarithm.
Help him answer the following questions!
a Use common logarithms and a calculator to find the value of log_4 23. Round the answer to two decimal places.
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b Use natural logarithms and a calculator to find the value of log_7 56. Round the answer to two decimal places.
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Hint
a By the Change of Base Formula, log_4 23= log 23log 4.
b By the Change of Base Formula, log_7 56= ln 56ln 7.
Solution
a Recall the Change of Base Formula.
log_b a=log_c a/log_c b Here, a, b, and c are positive numbers, with b and c different than 1. While the base c can be any number, in this case it might be most helpful if it was 10. Then the logarithm of 23 with base 4 could be expressed in terms of common logarithms. log_4 23=log 23/log 4 The value of the numeric expression on the right-hand side of the above equation can be found by using a calculator.
Therefore, the value of log_4 23 rounded to the nearest hundredth is 2.26.
b Once again, recall the Change of Base Formula, this time with natural logarithms.
ln_b a=log_c a/log_c b Here, a, b, and c are positive numbers, with b and c different than 1. For this exercise, let c be e. Then the logarithm of 56 with base 7 can be expressed in terms of natural logarithms. log_7 56=ln 56/ln 7 The value of the numeric expression on the right-hand side of the above equation can be found by using a calculator.
Therefore, the value of log_7 56 rounded to the nearest hundredth is 2.07.
Closure
Leonhard Euler and His Contributions
In this lesson, the Swiss mathematician Leonhard Euler was mentioned. Euler was also a physicist, astronomer, geographer, logician, and engineer. During his life, Euler came up with principles that set the foundations for most of the mathematics used nowadays. He was a revolutionary thinker in the fields of geometry, calculus, trigonometry, differential equations, and number theory.
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