Problem 14 - Precalculus Mathematics for Calculus

Question

Evaluate the expression. $$\log _{3} 100-\log _{3} 18-\log _{3} 50$$

Step-by-Step Solution

Answer

The expression evaluates to -2.

Step 1: Understand the Properties of Logarithms

To simplify the expression, we need to use the properties of logarithms. The key property we'll use is the subtraction property of logarithms, which states that \(\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)\).

Step 2: Apply the Subtraction Property

First, apply the logarithm subtraction property to combine the first two logarithms in the expression: \(\log_{3} 100 - \log_{3} 18\). This becomes \(\log_{3} \left(\frac{100}{18}\right)\).

Step 3: Simplify the Fraction

Simplify the fraction \(\frac{100}{18}\) which is equal to \(\frac{50}{9}\) by dividing both numerator and denominator by 2.

Step 4: Combine the Remaining Terms

Next, apply the subtraction property again to \(\log_{3} \left(\frac{50}{9}\right) - \log_{3} 50\), which becomes \(\log_{3} \left(\frac{\frac{50}{9}}{50}\right)\).

Step 5: Simplify the Expression

Simplify the fraction \(\frac{\frac{50}{9}}{50}\). This simplifies to \(\frac{1}{9}\) since the \(50\) in the numerator and the denominator cancel each other out.

Step 6: Final Logarithmic Expression

Now, the expression is \(\log_{3} \left(\frac{1}{9}\right)\). We know that \(\frac{1}{9} = 3^{-2}\), thus \(\log_{3} 3^{-2} = -2\).

Key Concepts

Properties of Logarithms

Logarithms are incredibly useful when dealing with complex calculations involving multiplication, division, powers, and roots. They rely on specific properties that simplify these operations. Here are the main properties of logarithms you should be familiar with:

• Product Property: When multiplying numbers within the logarithm, we have \( \log_b (MN) = \log_b M + \log_b N \).
• Quotient Property: For division within the logarithm, it's expressed as \( \log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N \).
• Power Property: When a number within the logarithm is raised to a power, the property is \( \log_b (M^n) = n \cdot \log_b M \).

These properties allow us to "condense" or "expand" logarithmic expressions. It's like simplifying or factoring numbers but in the realm of logarithms. For subtraction as seen in the problem, the quotient property plays a major role.

Subtraction Property of Logarithms

The subtraction property of logarithms is an essential tool in simplifying expressions that involve multiple logarithmic terms. Understanding this concept can unravel complex equations into more manageable forms.

Consider the subtraction property formula:
\[\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)\]This property tells us that when we subtract two logarithms with the same base, we can rewrite the expression using a single logarithm with the base and the division of their arguments.

• This not only helps in solving logarithmic equations efficiently but also aids in understanding the ratio between values \(M\) and \(N\).
• In the original exercise, this property is used at multiple stages to combine terms and ultimately simplify the expression from two terms to a single term that can be further simplified.

Remember, the bases for both logarithms must be the same for this property to apply. This simplification leads to more straightforward comparisons and calculations.

Simplifying Fractions

When working with logarithms, simplifying fractions within them is a common task. This stems from the subtraction property transforming logarithms into divisions, which often leads to fractions.

Let's look at how you can simplify fractions efficiently:

• Identify common factors in both the numerator and denominator. Divide the top and bottom by this common factor to reduce the fraction.
• Make use of prime factorization when necessary, which breaks numbers into their basic building blocks, making common factors easy to spot.

For example, with \(\frac{100}{18}\), both numbers are divisible by 2, simplifying it to \(\frac{50}{9}\). Further simplification isn't possible as 50 and 9 have no common factors other than 1.

Once combined using logarithm properties, a new fraction \(\frac{\frac{50}{9}}{50}\) may arise, which simplifies further by canceling terms, like exploring the expression: Just divide 50 in the numerator and 50 in the denominator leading into \(\frac{1}{9}\). Always look for opportunities to simplify; it makes the math more clear and the final logarithmic evaluation more manageable.