Evaluate logarithms

Learning Objectives Evaluate basic logarithms by converting them to their equivalent exponential form. Apply the Change of Base Formula to evaluate any logarithm using a calculator. Instantly evaluate special cases, including log_b(1), log_b(b), and log_b(b^x). Differentiate between and evaluate common logarithms (base 10) and natural logarithms (base e). Evaluate logarithms that result in integer, fractional, and negative answers. Set up and solve for an unknown variable in a simple logarithmic equation by means of evaluation. How many times more intense is a 7.0 magnitude earthquake than a 5.0? 🤯 The answer lies in logarithms, the powerful tool for taming gigantic numbers! Logarithms are the inverse of exponential functions; they essentially 'undo' exponentiati...

TermDefinitionExample LogarithmThe exponent to which a specified base must be raised to obtain a given number. The expression log_b(x) asks the question: 'What exponent do I put on base b to get the number x?'In log₂(8) = 3, the logarithm is 3 because 2³ = 8. Base (b)In a logarithm log_b(x), the base 'b' is the number being raised to a power. It must be a positive number and not equal to 1.In log₅(25), the base is 5. Argument (x)In a logarithm log_b(x), the argument 'x' is the number you are taking the logarithm of. It must be a positive number.In log₃(81), the argument is 81. Logarithmic Form vs. Exponential FormTwo equivalent ways to express the same relationship. The logarithmic form is log_b(x) = y, and the exponential form is b^y = x.log₂(16) = 4 is the...

Logarithmic-Exponential Equivalence log_b(x) = y \iff b^y = x This is the fundamental definition of a logarithm. Use it to convert a logarithm into an exponential equation, which is often easier to solve mentally or algebraically. Change of Base Formula log_b(a) = \frac{\log_c(a)}{\log_c(b)} This formula allows you to evaluate a logarithm of any base 'b' by converting it into a ratio of logarithms with a different base 'c'. It is essential for calculator use, where 'c' is typically 10 (log) or e (ln). Fundamental Logarithmic Identities 1. log_b(1) = 0 \quad 2. log_b(b) = 1 \quad 3. log_b(b^x) = x These are shortcuts for evaluation. Any base to the power of 0 is 1. Any base to the power of 1 is itself. The logarithm base b 'undoes&#0...