Quotient property of logarithms

Learning Objectives State the quotient property of logarithms from memory. Apply the quotient property to expand a single logarithm of a quotient into a difference of two logarithms. Apply the quotient property to condense a difference of two logarithms into a single logarithm of a quotient. Solve logarithmic equations by applying the quotient property to simplify the equation. Combine the quotient property with the product and power properties to manipulate complex logarithmic expressions. Evaluate numerical logarithmic expressions using the quotient property and given logarithm values. Differentiate between the quotient property, log(M/N), and the quotient of logs, log(M)/log(N). How do scientists compare the intensity of a massive earthquake to a tiny tremor? They use l...

TermDefinitionExample LogarithmThe exponent to which a specified base must be raised to obtain a given number. It answers the question: 'what exponent do I need?'log₂(8) = 3, because 2³ = 8. ArgumentThe value or expression inside the logarithm function.In log₃(x² - 1), the argument is x² - 1. BaseThe number that is being raised to a power in an exponential function, or the subscript in a logarithmic function.In log₅(25), the base is 5. QuotientThe result obtained by dividing one quantity by another.In the expression M/N, the entire expression represents a quotient. Expand a LogarithmTo rewrite a single logarithm with a complex argument as a sum, difference, or multiple of simpler logarithms.Expanding log(x/2) gives log(x) - log(2). Condense a LogarithmTo rewrite an expression co...

Quotient Property of Logarithms log_b(M/N) = log_b(M) - log_b(N) Use this rule to handle division inside a logarithm. To expand, break a single log of a fraction into the difference of two logs. To condense, combine a difference of two logs (with the same base) into a single log of a fraction. Product Property of Logarithms log_b(M * N) = log_b(M) + log_b(N) A related property used for multiplication inside a logarithm. It is often used in combination with the quotient property. Power Property of Logarithms log_b(M^p) = p * log_b(M) Used to handle exponents inside a logarithm. The exponent on the argument can be moved to the front as a coefficient. This is also frequently used with the quotient property.