2. Common and Natural Logarithms
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Chapter 5
2.
Common and Natural Logarithms
This lesson focuses on the mathematical concepts of exponents, common logarithms, and natural logarithms. It explains how these elements are essential tools for simplifying complex calculations in various fields like engineering, finance, and science. For instance, natural logarithms, which use a special number known as the natural base, are particularly useful in calculating compound interest or understanding exponential growth and decay. Common logarithms, on the other hand, are often employed in tasks like earthquake measurement and audio processing. By understanding these concepts, you can gain a deeper insight into the mathematical principles that govern various phenomena and make problem-solving more efficient.
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| | 14 Theory slides |
| | 14 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Common and Natural Logarithms
Slide of 14
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.
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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.
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.
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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.
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Evaluating Common Logarithms
Use a calculator to evaluate the common logarithms. Round the answer to two decimal places.
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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.
If two earthquakes have intensity levels I_1 and I_2, and magnitudes on the Richter scale M_1 and M_2, the following formula holds true. log I_1/I_2=M_1-M_2 In 1989, the city of San Francisco was jolted by an earthquake of magnitude 7.1 on the Richter scale. In 1906, nearly 80 years prior, the city had suffered a more powerful earthquake that is estimated to have measured magnitude 7.9 on the same scale. Vincenzo wants to know how many times more intense the 1906 earthquake was than to the 1989 earthquake.
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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.
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
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.
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Example
Using the Properties of Common Logarithms
After studying the relationship between earthquakes and logarithms, Vincenzo became more interested in this fascinating math topic.
Now he wants to pair the logarithmic expressions that involve common logarithms with their corresponding simplified expression or number. Help him do this!
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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
The expression log 10x - log x should be paired with 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
The expression 2log x+log 10x should be paired with 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
The expression log 0.1x should be paired with 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
The expression log 100x+log x^2 should be paired with 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
The expression log x+log x^2-3log x should be paired with 0.
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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.
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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.
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...
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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.
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Evaluating Natural Logarithms
Use a calculator to evaluate the natural logarithms. Round the answer to two decimal places.
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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.
{"type":"choice","form":{"alts":["ln x^2 +ln y\/x+ln z","ln ex+ln y^2z-(ln y+1)","- ln e\/xyz+1","ln xy+ln z-ln y","2ln sqrt(xy) +ln z^2 -ln z"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":3}
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
The expression ln x^2 +ln yx+ln z is equivalent to 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
The expression ln ex+ln y^2z-(ln y+1) is also equivalent to 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
The expression - ln exyz+1 is also equivalent to 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
The expression ln xy+ln z-ln y is equivalent to ln xz. Therefore, this expression is not equivalent to the previous expressions.
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
The last expression, 2ln sqrt(xy) +ln z^2 -ln z, is also equivalent to ln xyz. Therefore, the expression that is not equivalent to the others is the fourth expression.
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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 c
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Example
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.
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.
File:A2 U5 C2 Example4 2.svg
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.
File:A2 U5 C2 Example4 3.svg
Therefore, the value of log_7 56 rounded to the nearest hundredth is 2.07.
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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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Common and Natural Logarithms
Exercise 3.1
Consider the following inequality. ln e^x > 1 Determine whether this statement is sometimes, always, or never true.
{"type":"choice","form":{"alts":["Sometimes true","Always true","Never true"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
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To determine whether the inequality is sometimes, always, or never true, let's start by recalling the Power Property of Logarithms for a natural logarithm. ln a^m=mln a Also, since a natural logarithm is a logarithm with base e, we know that ln e=1. Let's use this information to simplify the left-hand side of the given inequality.
The given inequality is equivalent to x>1. This statement is true for values of x greater than 1 and false for values of x less than or equal to 1. Therefore, the statement x>1 is sometimes true and, therefore, the given inequality is also sometimes true.
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Solution
Consider the following equation. 10^(log x) = x Determine whether this statement is sometimes, always, or never true if x is any real number.
{"type":"choice","form":{"alts":["Sometimes true","Always true","Never true"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
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Let's use the definition of a common logarithm — a logarithm with base 10 — to rewrite the given equation. Definition:& 10^c= a ⇔ log a= c Equation:& 10^(log x)= x ⇔ log x= log x We found an equivalent equation to the given equation. log x = log x Since the expressions on both sides are the same, the equation seems to be always true. However, we need to consider all real numbers. If x is not a positive number, then log x is undefined. log x is not defined for x≤ 0 In this case, the statement is false because both sides are undefined. Conversely, if x is positive, both sides are defined and the statement holds true. Therefore, the statement is sometimes true.
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Common and Natural Logarithms
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