Power property of logarithms

Learning Objectives State the power property of logarithms and explain its derivation. Apply the power property to expand a single logarithm with a power into a product. Apply the power property to condense a product involving a logarithm into a single logarithm. Solve exponential equations by taking the logarithm of both sides and applying the power property. Solve logarithmic equations that require the use of the power property to simplify and isolate the variable. Simplify complex logarithmic expressions for differentiation by applying the power property first. Distinguish between log_b(M^p) and (log_b(M))^p and apply the power property correctly. How can you solve for 'x' in an equation like 7^x = 150 without just guessing? The power property of logarithms is...

TermDefinitionExample LogarithmA logarithm is the exponent to which a specified base must be raised to obtain a given number. It is the inverse operation of exponentiation.log₂(8) = 3, because 2³ = 8. BaseIn a logarithm log_b(x), the base 'b' is the number being raised to a power. Common bases are 10 (common log) and 'e' (natural log).In log₁₀(100), the base is 10. ArgumentThe number or expression inside the logarithm on which the function is operating.In ln(x²), the argument is x². Power Property of LogarithmsA property that states the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number.log(5³) is equivalent to 3 * log(5). Exponential EquationAn equation in which the variable appears in the exponent.4^(x+1) = 64 Loga...

Power Property of Logarithms log_b(M^p) = p * log_b(M) Use this rule to move an exponent from the argument of a logarithm to become a coefficient in front of the logarithm. This is crucial for solving for variables in exponents and for simplifying expressions. This works for any base b > 0, b ≠ 1, and for M > 0. Product Property of Logarithms log_b(M * N) = log_b(M) + log_b(N) Used to expand the logarithm of a product into a sum of logarithms. It is often used in conjunction with the power property. Quotient Property of Logarithms log_b(M / N) = log_b(M) - log_b(N) Used to expand the logarithm of a quotient into a difference of logarithms. It is also frequently used with the power property.