Mathematics
Grade 11
15 min
Quotient property of logarithms
Quotient property of logarithms
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1
Introduction & Learning Objectives
Learning Objectives
State the quotient property of logarithms from memory.
Apply the quotient property to expand a single logarithm with a quotient into a difference of two logarithms.
Apply the quotient property to condense a difference of two logarithms into a single logarithm.
Use the quotient property to simplify and evaluate numerical logarithmic expressions.
Use the quotient property as a key step in solving logarithmic equations.
Differentiate between the quotient property and the product property of logarithms.
Identify and avoid common errors when applying the quotient property.
How can you simplify the logarithm of a fraction like log(1000/10) without a calculator? 🤔 The quotient property turns this division problem into simple subtraction!
This tutorial focuse...
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Key Concepts & Vocabulary
TermDefinitionExample
LogarithmAn exponent that a specified base must be raised to in order to get a certain number. In `log_b(x) = y`, `y` is the logarithm.`log_2(8) = 3` because `2^3 = 8`.
BaseThe number that is being raised to a power. In `log_b(x)`, `b` is the base.In `log_5(25)`, the base is 5.
ArgumentThe value or expression inside the logarithm, which we are taking the logarithm of.In `log_3(x/2)`, the argument is `x/2`.
QuotientThe result obtained by dividing one quantity by another.In the expression `M/N`, the entire expression is the quotient.
Expand a LogarithmTo rewrite a single logarithm as a sum or difference of multiple, simpler logarithms.Expanding `log(a/b)` gives `log(a) - log(b)`.
Condense a LogarithmTo rewrite a sum or difference of logarithms (with the same base) as a...
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Core Formulas
Quotient Property of Logarithms
`log_b(M/N) = log_b(M) - log_b(N)`
Use this rule to handle division within a logarithm. The logarithm of a quotient is the difference of the logarithms of the numerator (M) and the denominator (N). This applies only when the bases are the same and M and N are positive.
Definition of a Logarithm
`log_b(x) = y` is equivalent to `b^y = x`
This fundamental rule allows you to convert between logarithmic and exponential forms. It is often used as the final step when solving a logarithmic equation after applying other properties.
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Challenging
Solve for x: log_4(x + 10) - log_4(x - 5) = 2. Be sure to check for extraneous solutions.
A.x = 5
B.x = 6
C.x = -6
D.No solution
Challenging
A student is presented with the following flawed proof: Step 1: `log_2(4) = 2`. Step 2: `4` can be written as `8 - 4`, so `log_2(8 - 4) = 2`. Step 3: Using a logarithm property, `log_2(8) - log_2(4) = 2`. Step 4: `3 - 2 = 2`. Step 5: `1 = 2`. In which step did the student first make a mistake?
A.Step 2
B.Step 3
C.Step 4
D.Step 5
Challenging
If log_a(x) - log_a(y) = log_a(y) - log_a(z), which of the following statements must be true?
A.x, y, and z form an arithmetic sequence.
B.x = z
C.2y = x + z
D.y^2 = xz
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