The change of base formula 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 $e$). The formula is: $\log_b a = \frac{\log_c a}{\log_c b}$, where $a$ is the argument, $b$ is the original base, and $c$ is the new base (usually 10 or $e$). This is super useful when your calculator can't directly compute a logarithm with a specific base!
For example, to evaluate $\log_5 16$, you can use the change of base formula to rewrite it as $\frac{\log_{10} 16}{\log_{10} 5}$ or $\frac{\ln 16}{\ln 5}$. Both expressions will give you the same result, which you can easily calculate using a calculator.
Match the term with its definition:
| Term | Definition |
|---|---|
| 1. Logarithm | A. The base that is commonly used for logarithms. |
| 2. Base | B. The inverse operation to exponentiation. |
| 3. Argument | C. The value inside the logarithm. |
| 4. Common Logarithm | D. The base to which we are raising a power. |
| 5. Natural Logarithm | E. Logarithm with base $e$. |
The change of base formula is $\log_b a = \frac{\log_c a}{\log_c b}$. This allows us to evaluate logarithms with any ____ (1) using a ____ (2). The most common new bases are ____ (3) and ____ (4), which are available on most calculators. Therefore, $\log_7 12$ can be rewritten as $\frac{\log 12}{\log 7}$ or $\frac{\ln 12}{\ln 7}$, where 'log' implies base ____ (5).
Explain why the change of base formula is useful when using a calculator. Give an example of a situation where it would be necessary to use this formula.
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