The change of base formula is a handy tool that allows you to evaluate logarithms with any base using a calculator, which typically only has buttons for common logarithms (base 10) and natural logarithms (base $e$). It's especially useful when dealing with logarithms that don't have straightforward integer or fractional answers. The formula essentially rewrites a logarithm with one base in terms of logarithms with a different base, enabling easy calculation. Understanding this formula is key to solving a variety of logarithmic problems in pre-calculus.
The formula is given by:
$\log_b a = \frac{\log_c a}{\log_c b}$
Where $a$ and $b$ are positive real numbers, $b \neq 1$, and $c$ is any positive real number different from 1.
Match the term to its definition:
| Term | Definition |
|---|---|
| 1. Logarithm | A. The base that is commonly used is 10. |
| 2. Base | B. The inverse operation to exponentiation. |
| 3. Argument | C. The number being input into the logarithm. |
| 4. Common Logarithm | D. The number that is raised to a power. |
| 5. Natural Logarithm | E. The base that is equal to $e$ ($2.71828...$). |
The change of base formula allows us to rewrite a logarithm with any ______ in terms of logarithms with a different ______. This is especially useful when evaluating logarithms on a ______ because they usually only have buttons for base ______ (common logarithm) and base ______ (natural logarithm).
Explain why it is important that the new base ($c$) in the change of base formula cannot be equal to 1.
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