Product property of logarithms

Learning Objectives State the product property of logarithms in its general form. Use the product property to expand a single logarithm of a product into a sum of two or more logarithms. Use the product property to condense a sum of logarithms (with the same base) into a single logarithm. Evaluate numerical logarithmic expressions by applying the product property. Solve simple logarithmic equations that require the use of the product property. Differentiate the product property from common incorrect operations, such as distributing a logarithm over a sum. Explain the connection between the product property of logarithms and the product rule for exponents. Ever wondered how scientists handle gigantic numbers in astronomy or chemistry? 🔭 They use a mathematical 'shortc...

TermDefinitionExample LogarithmAn exponent to which a specified base must be raised to obtain a given number. In the equation y = log_b(x), y is the logarithm, b is the base, and x is the argument.log_2(8) = 3, because 2^3 = 8. BaseThe number that is being raised to a power in an exponential expression, or the reference number for a logarithm.In log_5(25), the base is 5. ArgumentThe value or expression inside the logarithm function.In log_3(9x), the argument is 9x. Expand a LogarithmTo rewrite a single logarithm with a complex argument as a sum or difference of simpler logarithms.Expanding log_2(4x) gives log_2(4) + log_2(x). Condense a LogarithmTo rewrite a sum or difference of multiple logarithms (with the same base) as a single logarithm.Condensing log(5) + log(x) gives log(5x). Common...

Product Property of Logarithms log_b(M * N) = log_b(M) + log_b(N) Use this rule when you need to expand a single logarithm whose argument is a product, or when you need to condense a sum of two logarithms with the same base into a single logarithm. This rule requires that M > 0, N > 0, and b > 0, b ≠ 1. Connection to Exponent Rules b^x * b^y = b^(x+y) This is the product rule for exponents. The product property of logarithms is the direct inverse of this rule. Notice how multiplication of the main terms (b^x * b^y) corresponds to the addition of the exponents (x+y), just as multiplication of the arguments (M*N) corresponds to the addition of the logarithms.