- Identify the logarithmic expression \(\log_{5} 856\).
- Use the change-of-base formula \(\log_{5} 856 = \frac{\ln 856}{\ln 5}\).
- Compute the ratio to obtain the decimal value \(\approx 4.195421\).
- State the final result: \(\boxed{4.195421}\).
Explanation
- Analysis of the Logarithmic Expression
The objective is to evaluate the expression \(\log_{5} 856\). This is a logarithmic expression where the base is \(5\) and the argument is \(856\). A logarithm \(\log_{b} x = y\) is defined such that \(b^y = x\). Since \(5^4 = 625\) and \(5^5 = 3125\), we expect the result to be a value between \(4\) and \(5\).
- Applying the Change-of-Base Formula
To evaluate the logarithm with base \(5\), we use the change-of-base formula, which allows us to convert a logarithm into a ratio of natural logarithms or common logarithms. The formula is given by:
\[\log_{b} x = \frac{\ln x}{\ln b}\]
Applying this to our expression:
\[\log_{5} 856 = \frac{\ln 856}{\ln 5}\]
- Calculation and Rounding
Using the values for the natural logarithms, \(\ln 856 \approx 6.752269\) and \(\ln 5 \approx 1.609438\). Dividing these values:
\[\frac{6.752269}{1.609438} \approx 4.195421\]
Rounding the result to six decimal places, we obtain \(4.195421\).
Examples
Logarithms are essential in many scientific and financial fields. For example, if you invest money at a specific annual interest rate, you can use logarithmic functions to calculate how long it will take for your investment to reach a certain goal. If you have \(\$500\) that grows to \(\$856\) at an interest rate that compounds in a way related to base 5 growth, calculating the time needed involves evaluating \(\log_{5}(\frac{856}{500})\). This skill is foundational for modeling growth, decay, and complex patterns in data.