Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. log_√19 5

Verified step by step guidance

1. Identify the logarithm you need to approximate: \(\log_{\sqrt{19}} 5\).
2. Recall the change-of-base formula: \(\log_a b = \frac{\log_c b}{\log_c a}\), where \(c\) is any positive number (commonly 10 or \(e\)).
3. Apply the change-of-base formula using base 10 (common logarithm): \(\log_{\sqrt{19}} 5 = \frac{\log_{10} 5}{\log_{10} \sqrt{19}}\).
4. Express \(\sqrt{19}\) as \$19^{1/2}\( to simplify the denominator: \)\log_{10} \sqrt{19} = \log_{10} 19^{1/2} = \frac{1}{2} \log_{10} 19$.
5. Calculate the values of \(\log_{10} 5\) and \(\log_{10} 19\), then substitute back into the fraction and simplify to find the approximate value to four decimal places.

Key Concepts

Change-of-Base Theorem

The change-of-base theorem allows you to rewrite a logarithm with any base as a ratio of logarithms with a new base, typically base 10 or e. It states that log_b(a) = log_c(a) / log_c(b), where c is the new base. This is useful for calculating logarithms on calculators that only have log base 10 or natural log functions.

Logarithms with Radical Bases

When the base of a logarithm is a radical, such as √19, it can be expressed as an exponent (19^(1/2)). This helps simplify the logarithm using exponent rules, making it easier to apply the change-of-base formula and calculate the value accurately.

Rounding and Approximation

After calculating the logarithm using the change-of-base formula, the result should be rounded to the specified decimal places, here four decimal places. Proper rounding ensures the answer meets the precision requirements and is presented clearly.