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Logarithms of Quotients
The logarithm of the ratio or quotient of two numbers is the difference of the logarithms and can be proven using the first law of exponents.
Learning Objective

Apply the quotient rule of logarithms to expand or condense logarithms
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 logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 10 10 10 = 103. More generally, if x = b^{y}, then y is the logarithm of x to base b, and is written y = log_{b}(x), so log_{10}(1000) = 3 .
Interactive Graph: Graph of Binary Logarithm
The graph of the logarithm with base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log_{2}(8) = 3, because 23 = 8. The graph gets arbitrarily close to the y axis, but does not meet or intersect it. Try changing the value of x.
The logarithm of a product is the sum of the logarithms of the factors:
Similarly, the logarithm of the ratio of two numbers is the difference of the logarithms.
To prove this, let m = log_{x}a.
Rewrite the above expression as an exponent. x^{m} = a (log_{x}a asks " x to what power is a ?" And the equation answers: "x to the m is a.")
Let n = log_{x}b. Thus, x^{n}=b.
If we replace a and b based on the previous equations, we get:
This can be further simplified to:
Which, using the first law of exponents, can be written as:
This is the key step. Therefore, it can be seen that the properties of logarithms come directly from the laws of exponents. Replacing m and n with what they were originally defined as results in this equation:
Hence, the previous problem has been proven.
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Key Term Reference
 base
 Appears in these related concepts: Logarithms of Powers, The Role of the Kidneys in AcidBase Balance, and Biology: DNA Structure and Replication
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 expression
 Appears in these related concepts: Simplifying, Multiplying, and Dividing, Expressions and Sets of Numbers, and Bacterial Transformation
 logarithm
 Appears in these related concepts: Solving Problems with Logarithmic Graphs, Changing Logarithmic Bases, and Graphs of Logarithmic Functions
 quotient
 Appears in these related concepts: Difference Quotients, Rational Coefficients, and Division and Factors
Sources
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Cite This Source
Source: Boundless. “Logarithms of Quotients.” Boundless Algebra. Boundless, 01 Jul. 2015. Retrieved 02 Jul. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/exponentsandlogarithms5/propertiesoflogarithmicfunctions38/logarithmsofquotients18311094/