Evaluate logarithms: mixed review

Learning Objectives Evaluate basic logarithms by converting them to exponential form. Apply the product, quotient, and power rules to simplify and evaluate complex logarithmic expressions. Evaluate common logarithms (base 10) and natural logarithms (base e) with and without a calculator. Recognize and apply the identity properties of logarithms, such as log_b(1) and log_b(b). Use the change of base formula to evaluate logarithms with any positive base. Solve multi-step problems that require combining multiple logarithm properties and evaluation techniques. How can we compare the massive energy of an earthquake with the gentle rustle of leaves? 🌍 Logarithms provide the scale to measure and understand our world, from the incredibly large to the infinitesimally small. This tu...

TermDefinitionExample LogarithmA logarithm is the exponent to which a specified base must be raised to get a certain number. It answers the question: 'What exponent do I need?'log₂(8) = 3, because 2³ = 8. Here, 3 is the logarithm. BaseIn the expression log_b(x), 'b' is the base. It is the number that is being raised to a power.In log₅(25), the base is 5. ArgumentIn the expression log_b(x), 'x' is the argument. It is the number we are taking the logarithm of.In log₃(81), the argument is 81. Common LogarithmA logarithm with base 10. It is usually written as log(x) without an explicit base.log(100) = 2, because 10² = 100. Natural LogarithmA logarithm with base 'e' (Euler's number, approximately 2.718). It is written as ln(x).ln(e²) = 2. Exponentia...

The Definition of a Logarithm log_b(y) = x <=> b^x = y This is the fundamental relationship between logarithms and exponents. Use it to convert between forms, which is often the first step in solving a problem. Logarithm Properties Product: log_b(MN) = log_b(M) + log_b(N) Quotient: log_b(M/N) = log_b(M) - log_b(N) Power: log_b(M^p) = p * log_b(M) These three properties are used to expand or condense logarithmic expressions, making them easier to evaluate. They only work when the logarithms have the same base. Change of Base Formula log_b(a) = log_c(a) / log_c(b) Use this formula to evaluate a logarithm with a base that your calculator doesn't have (like base 7 or 6). You can change it to a more common base, like base 10 (log) or base e (ln).