Problem 89 - A Graphical Approach to Precalculus with Limits

Question

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{5} 10$$

Step-by-Step Solution

Answer

\(\log_{5} 10 \approx 1.43067\).

Step 1: Understand the Change-of-Base Formula

The change-of-base formula allows you to express a logarithm in terms of logarithms with another base, usually a base that can be easily calculated, like 10 or e. The formula is \(\log_{b} a = \frac{\log_{c} a}{\log_{c} b},\) where \(c\) is a new base, often chosen as 10 or e for convenience.

Step 2: Apply the Change-of-Base Formula

To find \(\log_{5} 10\), we can set base \(c\) as 10. Thus, using the formula, we have: \[\log_{5} 10 = \frac{\log_{10} (10)}{\log_{10} (5)}.\]

Step 3: Calculate \(\log_{10} (10)\)

We know that \(\log_{10} (10) = 1\), because the logarithm with the same base as the argument is always 1.

Step 4: Approximate \(\log_{10} (5)\)

To approximate \(\log_{10} (5)\), we can use a calculator. Typically, \(\log_{10} (5)\) is approximately 0.69897.

Step 5: Compute the Expression

Substitute the known values into the expression: \[\log_{5} 10 = \frac{1}{0.69897} \]Calculate the division: \[\log_{5} 10 \approx 1.43067.\]

Key Concepts

Understanding Logarithms

Logarithms are mathematical entities that help you determine the power to which a number, called the base, must be raised to obtain another number. In simpler terms, if you have a logarithm like \(\log_b a\), it is asking, "What power must we raise \(b\) to get \(a\)?"

• The base in the logarithm determines the number from which we start.
• The result is the exponent or power you use with that base to produce the number in question.

Common logarithms (base 10) and natural logarithms (base \(e\)) are frequently used because they simplify calculations. Remember, logarithms have the inverse relationship with exponents, which means they undo what exponents do.

Approximating Values in Logarithms

Sometimes, logarithmic expressions do not give integer results, necessitating the use of approximation. Approximating helps when the precise value is difficult or impossible to find manually.

• For example, \(\log_{10} (5)\) is not easily derived without a calculator.
• In practice, calculators approximate values for logarithms that aren't whole numbers or powers of the base.
• Approximations can be highly precise, often providing enough accuracy for practical purposes.

When you approximate using a calculator, the result will be a decimal figure. Ensure you carry sufficient digits for the precision your task requires. In our exercise, \(\log_{10} (5)\) is approximately 0.69897, which is accurate for most needs.

Converting Bases in Logarithmic Expressions

Converting logarithmic bases makes certain calculations more manageable. The change-of-base formula shows how to express a logarithm in one base in terms of a logarithm with another base.This is especially useful when you have a calculator that only handles certain bases, such as 10 or \(e\). The formula\(\text{Change-of-base formula}: \log_b a = \frac{\log_c a}{\log_c b}\) is key.

• Choose a calculator-friendly base, often base 10 or \(e\).
• Substitute into the formula to find the new equivalent expression.

In the exercise, changing the base from 5 to 10 simplifies solving \(\log_5 10\) by converting it into a ratio of known logarithms: \(\frac{\log_{10} (10)}{\log_{10} (5)}\). This makes calculating more accessible, ultimately giving the approximate value 1.43067 for \(\log_5 10\).