Problem 107 - College Algebra

Question

Without using a calculator, determine which is the greater number: \(\log _{4} 60\) or \(\log _{3} 40.\)

Step-by-Step Solution

Answer

\(\log_3 40\) is greater than \(\log_4 60\).

Step 1: Rewrite with the change of base formula

First, rewrite the given logarithms using the change of base formula: \(log_b a > log_c d\) if and only if \(a^c > b^d\). This gives us \(60^3 > 4^{40}\)

Step 2: Break down expressions to compare them

Evaluate this expression involves large numbers which proves to be difficult without a calculator so it is best to break it down into smaller, more manageable parts which are easier to compare. To do this, apply the property of exponents \(b^{m*n} = (b^m)^n\). This gives \(60^3 = (4*15)^3 = 4^3 * 15^3\) and \(4^{40} = (4^{10})^4\).

Step 3: Comparison

Once the expression has been simplified, we see that we have \(4^3\) on one side and \(4^4\) on the other side, so clearly \(4^4\) is larger and therefore \(log_3 40\) is greater than \(log_4 60\).

Key Concepts

Understanding the Change of Base Formula

The change of base formula is a fundamental tool within logarithms that helps us compare logarithms with different bases. This formula allows us to express logarithms in terms of a convenient base, often base 10 or base 'e', which are more familiar and easier to work with.
For any logarithm \log_b(a)\, the change of base formula is expressed as:

• \(\log_b a = \frac{\log_k a}{\log_k b}\)

Here, \(k\) is the new base of your choice. Using this formula, you can change both \(\log_4(60)\) and \(\log_3(40)\) into common logarithms (base 10 or natural log, base 'e'), making it simpler to compare them directly.
This technique is crucial for solving logarithmic problems that involve comparisons or where a calculation might not seem straightforward at first glance.

Demystifying Exponents

Exponents, also known as powers, represent repeated multiplication. For instance, \(4^3\) means multiplying 4 by itself three times, which equals 64. Exponents have several important properties that are useful in various mathematical operations.

Key exponent rules include:

• Power of a Power: \(b^{m\cdot n} = (b^m)^n\)
• Product of Powers: \(b^m \cdot b^n = b^{m+n}\)
• Zero Exponent: \(b^0 = 1\) for any non-zero \(b\)
• Negative Exponent: \(b^{-n} = \frac{1}{b^n}\)

Understanding these rules allows you to simplify complex expressions, making it easier to compare values and derive solutions. In the context of comparing numbers like in the original exercise, the exponent relation helps break large calculations into simpler comparisons.

Logarithmic Comparison Made Simple

Logarithmic comparison involves determining which of two logarithmic expressions represents the greater value. To do this effectively, converting logarithmic form into exponential form can often make evaluation more intuitive.
In the original exercise, \(\log_4 60\) and \(\log_3 40\) were compared using their exponent forms. Using the property that \(a^c > b^d\) implies \(\log_b a > \log_c d\), we can transition from logarithms to exponents.

• Convert: Determine equivalent exponential expressions for each logarithm.
• Simplify: Break down into smaller components if the numbers are too large.
• Compare: Analyze the simplified expressions for clear understanding.

This method helps visualize which numerical value is greater without relying directly on logarithms for measurable quantities. Thus, ensuring that the larger exponent has the larger overall value based on its base raised to the compared power.