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Logarithms of Quotients
The logarithm of the ratio of two quantities is the difference of the logarithms of the quantities. In symbols,
Learning Objective

Relate the quotient rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of quotients
Key Points
Term

exponent
The power to which a number, symbol, or expression is to be raised. For example, the 3 in
$x^3$ .
Full Text
We have already seen that the logarithm of a product is the sum of the logarithms of the factors:
Similarly, the logarithm of the ratio of two quantities is the difference of the logarithms:
We can show that this is true by the following example:
Let
Then
Then:
Another way to show that this rule is true, is to apply both the power and product rules and the fact that dividing by
Example: write the expression $\log_2\left({x^4y^9 \over z^{100}}\right)$ in a simpler way
By applying the product, power, and quotient rules, you could write this expression as:
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Key Term Reference
 base
 Appears in these related concepts: Temple Architecture in the Greek Orientalizing Period, Rules for Exponent Arithmetic, and Rational Exponents
 difference
 Appears in these related concepts: Asymptotes, Graphs of Equations as Graphs of Solutions, and Factoring a Difference of Squares
 expression
 Appears in these related concepts: Compound Inequalities, Sets of Numbers, and Simplifying, Multiplying, and Dividing
 factor
 Appears in these related concepts: Rational Algebraic Expressions, Finding Factors of Polynomials, and Factors
 logarithm
 Appears in these related concepts: Logarithms of Powers, Changing Logarithmic Bases, and The Number e
 product
 Appears in these related concepts: Measuring Reaction Rates, Writing Chemical Equations, and Basic Operations
 quotient
 Appears in these related concepts: Division and Factors, Permutations of Distinguishable Objects, and Zeroes of Polynomial Functions With Rational Coefficients
 sum
 Appears in these related concepts: What Are Polynomials?, The Order of Operations, and Scientific Applications of Quadratic Functions
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Cite This Source
Source: Boundless. “Logarithms of Quotients.” Boundless Algebra. Boundless, 05 Oct. 2016. Retrieved 21 Oct. 2016 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/exponentslogarithmsandinversefunctions8/workingwithlogarithms355/logarithmsofquotients18311094/