Question

Use the change of base formula to rewrite the given expression in terms of natural logarithms or common logarithms

Answer 1

To rewrite logarithms using the change of base formula, use the rule log_b(x) = log_c(x) / log_c(b), which allows conversion to either natural logarithm (ln) or common logarithm (log) in calculations.

Step-by-step explanation:

The change of base formula allows us to rewrite logarithms in terms of natural logarithms (ln) or common logarithms (log). When working with exponential growth, we often use the number e, approximately equal to 2.7182818, which is the base of natural logarithms. If we want to express a base b raised to a power n, we can use the identities bn = en ln(b) = 10n log10(b).

For example, computing the natural logarithm of a number can be done using the ln button on a calculator, while the inverse operation is done with the exponent key, entering ex, where x is the natural logarithm. On the other hand, the common logarithm of a number is its exponentiation base 10, often used when base e isn't as convenient. The natural logarithm and the common logarithm are connected through a property where the natural logarithm of 10 is around 2.303.

When a calculator lacks a specific base logarithm function, the change of base formula is especially useful. It follows the rule logb(x) = logc(x) / logc(b), where c can be e (for ln) or 10 (for log). This enables computation of logarithms with any base by using the more commonly available ln or log functions.

Answer 2

Given:

`\log _{\sqrt[]{3}}20`

The change of base formula states that,

`\log _bx=(\log _cx)/(\log _cb)`

For the given expression,

`\begin{gathered} \log _{\sqrt[]{3}}20=\frac{\log_(10)20}{\log_(10)\sqrt[]{3}} \\ =(\log_(10)20)/((1)/(2)\log_(10)3) \\ =(2\log_(10)20)/(\log_(10)3) \end{gathered}`

Answer:

`\begin{gathered} \log _{\sqrt[]{3}}20=\frac{\log_(10)20}{\log_(10)\sqrt[]{3}} \\ or\text{ in simplified form,} \\ \log _{\sqrt[]{3}}20=(2\log_(10)20)/(\log_(10)3) \end{gathered}`