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| 11 Theory slides |
| 11 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
A logarithm is the inverse function of an exponential function. The logarithm of a positive number m is written as logbm and read as the logarithm of m with base b.
logbm=n⇔bn=m
As a consequence of the definition of a logarithm, two properties can be deduced. In these properties, b is positive and not equal to 1.
Property | Reason |
---|---|
logbb=1 | A number raised to the power of 1 is equal to itself. |
logb1=0 | A number raised to the power of 0 is equal to 1. |
Paulina has recently become excited learning about logarithms.
She eagerly went to her math teacher and asked for some introductory exercises to practice evaluating and rewriting logarithmic expressions. Help her get off to a good start!
Evaluate the logarithms.
Rewrite the logarithmic equations as exponential equations and the exponential equations as logarithmic equations.
logba=c ⇔ a=bc |
---|
a=bc Substitute a=blogba |
logba=c Substitute logbbc=c |
With this information in mind, three properties can be stated.
The logarithm of a product can be written as the sum of the individual logarithms of each factor.
logbmn=logbm+logbn
Rewrite mn as m⋅n
m=blogb(m)
am⋅an=am+n
logb(bm)=m
The logarithm of a quotient can be written as the difference between the logarithm of the numerator and the logarithm of the denominator.
logbnm=logbm−logbn
m=blogb(m)
anam=am−n
logb(bm)=m
The logarithm of a power can be written as the product of the exponent and the logarithm of the base.
logbmn=nlogbm
m=blogb(m)
(am)n=am⋅n
logb(bm)=m
Commutative Property of Multiplication
Split into factors
log2(mn)=log2(m)+log2(n)
log23≈1.585, log25≈2.322
Add terms
log2(ba)=log2(a)−log2(b)
log2(1)=0
log25≈2.322
Subtract term
Split into factors
log2(mn)=log2(m)+log2(n)
Write as a power
log2(am)=m⋅log2(a)
log2(2)=1
Identity Property of Multiplication
log23≈1.585
Add terms
log5(ba)=log5(a)−log5(b)
log5(mn)=log5(m)+log5(n)
log5(am)=m⋅log5(a)
Calculate logarithm
log4(am)=m⋅log4(a)
log4(ba)=log4(a)−log4(b)
log4(mn)=log4(m)+log4(n)
Distribute 21
log4(4)=1
Identity Property of Multiplication
m⋅log2(a)=log2(am)
Calculate power
log2(m)+log2(n)=log2(mn)
Multiply
log2(m)−log2(n)=log2(nm)
Calculate quotient
m⋅log3(a)=log3(am)
Calculate power
log3(m)−log3(n)=log3(nm)
log3(m)+log3(n)=log3(mn)
ca⋅b=ca⋅b
Definition | logba=c ⇔ a=bc |
---|---|
Identity Derived From the Definition | logbb=1 |
Identity Derived From the Definition | logb1=0 |
Product Property of Logarithms | logbmn=logbm+logbn |
Quotient Property of Logarithms | logbnm=logbm−logbn |
Power Property of Logarithms | logbmn=nlogbm |
It is important to keep in mind that these properties are only valid for positive values of a, b, m, and n, where b=1. Furthermore, these properties can be used in several situations.
Evaluate the logarithms.
Let's begin by recalling the definition of a logarithm. log_b a = c ⇔ b^c=a In order to solve the given logarithmic expression, we need to think about which number we need to raise 3 to in order to get 81. log_3 81=? ⇔ 3^? =81 To find the number we are looking for, we will make a table of values to calculate the value of different powers of 3.
Powers of 3 | |
---|---|
Exponent | Power |
0 | 3^0=1 |
1 | 3^1=3 |
2 | 3^2=9 |
3 | 3^3=27 |
4 | 3^4=81 |
Because 3 raised to the fourth power equals 81, we know that log_3 81 is equal to 4. 3^4 = 81 ⇔ log_3 81= 4
Since any number raised to the power of 1 is equal to itself, we know that 3 raised to the power of 1 equals 3. We can use this information and the definition of a logarithm to find the value of log_3 3. 3^1= 3 ⇔ log_3 3= 1
Any non-zero number raised to the power of 0 is equal to 1. As such, we know that 12 raised to the power of 0 equals 1. We can use this information and the definition of a logarithm to find the value of log_(12) 1. ( 1/2)^0= 1 ⇔ log_(12) 1= 0
Let's start by rewriting 0.25 as 14. log_4 0.25 ⇔ log_4 1/4 Next, recall the Negative Exponent Property. a^(- n)=1/a^n, fora≠ 0andnnatural Therefore, 4 raised to the power of - 1 is equal to 14^1, or 14. By using this information and the definition of a logarithm, we know that the value of log_4 14 is - 1. 4^(- 1)= 1/4 ⇔ log_4 1/4= - 1 This means that log_4 0.25=- 1.
Rewrite the logarithmic expressions as exponential expressions.
Let's start by recalling the definition of a logarithm. log_b a= c ⇔ b^c= a We can use this definition to rewrite the given logarithmic expression as an exponential expression. log_5 x= y ⇔ 5^y= x
As we did in Part A, let's start by recalling the definition of a logarithm.
log_b a= c ⇔ b^c= a
Now let's rewrite the given logarithmic expression as an exponential expression.
log_b 10= y ⇔ b^y= 10
Rewrite the exponential expressions as logarithmic expressions.
Let's start by recalling the definition of a logarithm. b^c= a ⇔ log_b a= c We can use this definition to rewrite the given exponential expression as a logarithmic expression. 3^m= n ⇔ log_3 n= m
Like we did in part A, let's start by recalling the definition of a logarithm.
b^c= a ⇔ log_b a= c
Now let's rewrite the given exponential expression as a logarithmic expression.
x^2= y ⇔ log_x y= 2
We can use the fact that log_7 4 ≈ 0.712 to approximate the value of the given logarithmic expression. To expand the expression, first recall the Power Property of Logarithms. log_b m^n = n log_b m In this formula, m, n, and b are positive numbers, where b≠ 1. We can now rewrite 16 as a power of 4, apply the property, and then substitute 0.712 for log_7 4.
We can use the fact that log_7 4 ≈ 0.712 again to approximate the value of the given logarithmic expression. To expand the expression, we will first recall the Quotient Property of Logarithms. log_b m/n=log_b m - log_b n In this formula, m, n, and b are positive numbers, where b≠ 1. Let's apply the property and then substitute 0.712 for log_7 4. Keep in mind that log_b 1=0.
We can use the given values to approximate the value of the given logarithmic expression. To expand the expression, we will first recall the Quotient Property of Logarithms. log_b m/n=log_b m - log_b n In this formula, m, n, and b are positive numbers, where b≠ 1. Next, we can find an equivalent fraction to 13 with denominator 12, apply the property, and then substitute 0.712 for log_7 4 and 1.277 for log_7 12.
Use the properties of logarithms to fully expand the logarithmic expressions.
To expand the given logarithmic expression, we will first use the Quotient Property of Logarithms and then the Product Property of Logarithms.
To expand the given logarithmic expression, we will use the Product Property of Logarithms, the Quotient Property of Logarithms, and then the Power Property of Logarithms.
To expand the given logarithmic expression, we will start by rewriting sqrt(x) as x^(12). Then, we will use the Product Property of Logarithms and the Power Property of Logarithms. Let's do it!
Use the properties of logarithms to fully condense the logarithmic expressions.
To write the given expression as a single logarithm, we will use the Power Property of Logarithms and the Quotient Property of Logarithms. Let's do it!
To write the given expression as a single logarithm, we will use the Power Property of Logarithms and the Product Property of Logarithms. Let's do it!