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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
 The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.
 A basic idea in logarithmic math is that the logarithm of a product is the sum of the logarithms of the factors.
 A similar idea of the law of products is that the logarithm of the ratio or quotient of two numbers is the difference of the logarithms.
Term

exponent
The power to which a number, symbol, or expression is to be raised. For example, the 3 in
$x^3$ .
Full Text
The quotient rule for logarithms
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 see that this is true by letting
Another way to see that this rule is true is to apply both the power and product rules, and the fact that dividing by
Example
Suppose you wanted to write the expression
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Key Term Reference
 Base
 Appears in these related concepts: Rules for Exponent Arithmetic, Negative Exponents, and Simplifying Exponential Expressions
 base
 Appears in these related concepts: Changing Logarithmic Bases, Logarithms of Powers, and Rational Exponents
 difference
 Appears in these related concepts: Functions and Their Notation, Asymptotes, and Factoring a Difference of Squares
 expression
 Appears in these related concepts: Compound Inequalities, Sets of Numbers, and Graphs of Equations as Graphs of Solutions
 factor
 Appears in these related concepts: Rational Algebraic Expressions, Finding Factors of Polynomials, and Factors
 logarithm
 Appears in these related concepts: Solving Problems with Logarithmic Graphs, Graphs of Logarithmic Functions, and The Number e
 product
 Appears in these related concepts: Writing Chemical Equations, Measuring Reaction Rates, and The State of Competition
 quotient
 Appears in these related concepts: Zeroes of Polynomial Functions With Rational Coefficients, Division and Factors, and Basic Operations
 sum
 Appears in these related concepts: The Order of Operations, What Are Polynomials?, and Scientific Applications of Quadratic Functions
Sources
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Cite This Source
Source: Boundless. “Logarithms of Quotients.” Boundless Algebra. Boundless, 06 Jul. 2016. Retrieved 30 Jul. 2016 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/exponentslogarithmsandinversefunctions8/workingwithlogarithms355/logarithmsofquotients18311094/