Problem 283 - Precalculus

Question

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places. $$ \log _{6}(5.38) $$

Step-by-Step Solution

Answer

The value of \(\log_{6}(5.38)\) is approximately 0.93848.

Step 1: Recall the Change-of-Base Formula

The change-of-base formula for logarithms states that for any positive numbers \(a\), \(b\), and \(c\) (where \(a eq 1\) and \(c eq 1\)), \(\log_{a}(b) = \frac{\log_{c}(b)}{\log_{c}(a)}\). We'll use this formula to convert the logarithm from base 6 to base e (natural logarithm).

Step 2: Apply the Change-of-Base Formula

Using the change-of-base formula with base \(e\), we have:\[\log_{6}(5.38) = \frac{\ln(5.38)}{\ln(6)}\] This transforms the expression into a form that can be evaluated using a calculator since most calculators provide the natural logarithm function (ln).

Step 3: Calculate Natural Logarithms

Using a calculator, find the natural logarithms:\(\ln(5.38) \approx 1.68124\)\(\ln(6) \approx 1.79176\)Ensure your calculator is operating in the proper mode for computing natural logarithms (usually just 'ln' and no special mode is needed).

Step 4: Evaluate the Quotient

Now substitute the values obtained in Step 3 into the quotient:\[\log_{6}(5.38) = \frac{1.68124}{1.79176} \approx 0.93848\]Use the approximate values from the calculator to five decimal places for accuracy.

Key Concepts

Exploring Logarithms

Logarithms are mathematical expressions that help us solve equations involving exponential growth or decay. They tell us the power to which a number, known as the base, must be raised to produce another number.
For instance, in the expression \(\log_{b}(x)\), \(b\) is the base, and the logarithm answers the question: "To what power should \(b\) be raised, to get \(x\)?" For example, \(\log_{10}(100)\) equals 2, because you need to raise 10 to the power of 2 to get 100.
Understanding logarithms is crucial since they appear frequently in diverse fields like science, business, and computing. They simplify complex multiplicative processes into simpler additive ones, making calculations more manageable.

The Natural Logarithm Explained

The natural logarithm is a specific type of logarithm that uses the number \(e\) as its base. \(e\) is approximately equal to 2.71828 and is a fundamental constant in mathematics, much like \(\pi\). It is denoted by 'ln', short for "logarithmus naturalis" in Latin.
Natural logarithms are crucial in calculus and exponential growth scenarios, such as population models or financial calculations involving compound interest. Using natural logarithms can simplify many problems that involve exponential functions.
In real-life applications, natural logarithms are useful for solving problems relating to growth and decay, such as calculating the time needed for an investment to grow at a certain rate. They allow us to easily handle equations that involve continuous growth.

Calculator Approximation Made Easy

One of the benefits of the change-of-base formula is that it converts logarithms to a form that can be easily calculated using a standard calculator. Most modern calculators possess functions to calculate both common and natural logarithms, usually represented by 'log' and 'ln' buttons.
For evaluating expressions using the change-of-base formula, follow these steps:

• Use the 'ln' button to compute the natural logarithm of both the number and the base.
• Next, calculate the quotient of these two natural logarithms.
• Finally, round your answer to the desired precision, often five decimal places for accuracy.

This method simplifies solving logarithmic expressions, ensuring you can tackle even complex problems with ease. Always double-check your calculator results to prevent and correct any entry errors.