For a certain constant a>1, ln a≈3.8067 . Find approximate values of  log₂ a and logₐ 2 using the fact that ln 2≈0.6931.

Verified step by step guidance

1. Use the change of base formula for logarithms: ${\log }_{b}a=\frac{\ln a}{\ln b}$.
2. To find ${\log }_{2}a$, apply the formula: ${\log }_{2}a=\frac{\ln a}{\ln 2}$.
3. Substitute the given values: $\ln a\approx 3.8067$ and $\ln 2\approx 0.6931$.
4. Calculate ${\log }_{2}a$ using the substituted values: ${\log }_{2}a=\frac{3.8067}{0.6931}$.
5. To find ${\log }_{a}2$, use the reciprocal property of logarithms: ${\log }_{a}2=\frac{1}{{\log }_{2}a}$.

Key Concepts

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another. It states that log_b(x) = log_k(x) / log_k(b) for any positive k. This is particularly useful when we need to compute logarithms in a base that is not readily available, such as converting natural logarithms (ln) to base 2 or base a.

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is a fundamental concept in calculus and is used extensively in various applications, including growth models and compound interest. In this problem, ln a and ln 2 are provided, which are essential for calculating log₂ a and logₐ 2.

Approximation of Logarithmic Values

In this context, approximating logarithmic values involves using known logarithmic values to estimate others. Given ln a and ln 2, we can derive log₂ a and logₐ 2 using the change of base formula. This method is particularly useful when exact values are difficult to compute, allowing for practical estimations in mathematical problems.