Change of base formula

Learning Objectives State the change of base formula from memory. Use a scientific calculator to evaluate any logarithm by converting it to base 10 (common log) or base e (natural log). Rewrite a logarithmic expression from one base to another specified base. Simplify complex logarithmic expressions involving multiplication or division of logs with different bases. Solve logarithmic equations where the terms have different bases by applying the change of base formula. Explain why the change of base formula is a necessary tool for practical calculations. Ever tried to calculate log₃(10) on your calculator and found only 'LOG' and 'LN' buttons? 🤔 The change of base formula is the secret key that unlocks every other base! This tutorial will introduce you t...

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. BaseIn a logarithm logₐ(x), the base 'a' is the number that is being raised to a power. It is the foundation of the logarithmic system.In log₅(25), the base is 5. ArgumentIn a logarithm logₐ(x), the argument 'x' is the number you are taking the logarithm of.In log₅(25), the argument is 25. Common LogarithmA logarithm with base 10. On calculators, this is the 'LOG' button. If no base is written, it is assumed to be 10.log(1000) is the same as log₁₀(1000), which equals 3. Natural LogarithmA logarithm with base 'e' (Euler's number, appr...

General Change of Base Formula log_b(x) = \frac{log_c(x)}{log_c(b)} This formula allows you to convert a logarithm with an 'old' base 'b' to any 'new' base 'c'. The argument of the original log (x) goes in the numerator, and the original base (b) becomes the argument in the denominator. Change to Common Log (Base 10) log_b(x) = \frac{log(x)}{log(b)} This is a specific application of the general formula, where the new base is 10. This is the most common version used for scientific calculators. Change to Natural Log (Base e) log_b(x) = \frac{ln(x)}{ln(b)} This is another specific application, where the new base is 'e'. This is frequently used in calculus and higher-level mathematics.