This is “Properties of the Logarithm”, section 7.4 from the book Advanced Algebra (v. 1.0). For details on it (including licensing), click here.
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Recall the definition of the base-b logarithm: given where ,
Use this definition to convert logarithms to exponential form. Doing this, we can derive a few properties:
Evaluate:
Solution:
When the base is not written, it is assumed to be 10. This is the common logarithm,
The natural logarithm, by definition, has base e,
Because we have,
Furthermore, consider fractional bases of the form where
Evaluate:
Solution:
Given an exponential function defined by , where and , its inverse is the base-b logarithm, And because and , we have the following inverse properties of the logarithmGiven we have and when :
Since has a domain consisting of positive values , the property is restricted to values where
Evaluate:
Solution:
Apply the inverse properties of the logarithm.
In summary, when and , we have the following properties:
, |
In this section, three very important properties of the logarithm are developed. These properties will allow us to expand our ability to solve many more equations. We begin by assigning u and v to the following logarithms and then write them in exponential form:
Substitute and into the logarithm of a product and the logarithm of a quotient Then simplify using the rules of exponents and the inverse properties of the logarithm.
Logarithm of a Product |
Logarithm of a Quotient |
---|---|
This gives us two essential properties: the product property of logarithms the logarithm of a product is equal to the sum of the logarithm of the factors.,
and the quotient property of logarithms the logarithm of a quotient is equal to the difference of the logarithm of the numerator and the logarithm of the denominator.,
In words, the logarithm of a product is equal to the sum of the logarithm of the factors. Similarly, the logarithm of a quotient is equal to the difference of the logarithm of the numerator and the logarithm of the denominator.
Write as a sum:
Solution:
Apply the product property of logarithms and then simplify.
Answer:
Write as a difference: .
Solution:
Apply the quotient property of logarithms and then simplify.
Answer:
Next we begin with and rewrite it in exponential form. After raising both sides to the nth power, convert back to logarithmic form, and then back substitute.
This leads us to the power property of logarithms; the logarithm of a quantity raised to a power is equal to that power times the logarithm of the quantity.,
In words, the logarithm of a quantity raised to a power is equal to that power times the logarithm of the quantity.
Write as a product:
Solution:
Apply the power property of logarithms.
Recall that a square root can be expressed using rational exponents, Make this replacement and then apply the power property of logarithms.
In summary,
Product property of logarithms |
|
Quotient property of logarithms |
|
Power property of logarithms |
We can use these properties to expand logarithms involving products, quotients, and powers using sums, differences and coefficients. A logarithmic expression is completely expanded when the properties of the logarithm can no further be applied.
It is important to point out the following:
Expand completely:
Solution:
Recall that the natural logarithm is a logarithm base e, Therefore, all of the properties of the logarithm apply.
Answer:
Expand completely:
Solution:
Begin by rewriting the cube root using the rational exponent and then apply the properties of the logarithm.
Answer:
Expand completely: .
Solution:
When applying the product property to the denominator, take care to distribute the negative obtained from applying the quotient property.
Answer:
Caution: There is no rule that allows us to expand the logarithm of a sum or difference. In other words,
Given that , , and that , write the following in terms of a, b and c:
a.
b.
Solution:
Begin by expanding using sums and coefficients and then replace a and b with the appropriate logarithm.
Expand and then replace a, b, and c where appropriate.
Next we will condense logarithmic expressions. As we will see, it is important to be able to combine an expression involving logarithms into a single logarithm with coefficient 1. This will be one of the first steps when solving logarithmic equations.
Write as a single logarithm with coefficient
Solution:
Begin by rewriting all of the logarithmic terms with coefficient 1. Use the power rule to do this. Then use the product and quotient rules to simplify further.
Answer:
Write as a single logarithm with coefficient
Solution:
Begin by writing the coefficients of the logarithms as powers of their argument, after which we will apply the quotient rule twice working from left to right.
Answer:
Evaluate:
Find a:
Expand completely.
Given , , and , write the following logarithms in terms of a, b, and c.
Given , , and , calculate the following. (Hint: Expand using sums, differences, and quotients of the factors 2, 3, and 7.)
Expand using the properties of the logarithm and then approximate using a calculator to the nearest tenth.
Write as a single logarithm with coefficient 1.
Express as a single logarithm and simplify.
0
14
10
7
1
−1
100
18
4
2.32
0.71