Problem 69 - Precalculus

Question

Simplify if possible and explain the mistake that is made. \(\log _{2} x+\log _{3} y-\log _{4} z\) Solution: Apply the product property (5) $$\log _{6} x y-\log _{4} z$$ Apply the quotient property (6) $$\log _{24} x y z$$ This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The logs can't be combined due to different bases.

Step 1: Understand Logarithmic Properties

We need to recall the properties of logarithms. The product property states that \(\log_b (MN) = \log_b M + \log_b N\) and the quotient property states that \(\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N\). Both require the same base for operations.

Step 2: Identify the Mistake

In the given solution, the student combined logs with different bases using the product property. However, combining logs from different bases is not allowed.

Step 3: Reanalyze Each Logarithm

Analyzing \(\log_2 x + \log_3 y - \log_4 z\), none of these logarithms share the same base. Therefore, upon checking each term, it is clear that operations like combining or simplifying across logs with different bases can't be correctly performed using the mentioned properties.

Step 4: Conclusion

The primary mistake in the solution was attempting to combine logarithms with different bases using properties that require the same base. The expression \(\log_2 x + \log_3 y - \log_4 z\) cannot be simplified further in its current form due to differing bases.

Key Concepts

Logarithmic Properties

When working with logarithms, it's important to understand their foundational properties, as these rules help simplify expressions and solve equations. The most essential properties of logarithms are:

• Product Property
• Quotient Property
• Power Property

Each of these properties assumes that the logarithms have the same base. If the bases differ, as they do in the expression \( \log_2 x + \log_3 y - \log_4 z \), these properties cannot directly combine them.
Understanding these properties allows for proper manipulation of logarithmic expressions. However, as each mathematical property has specific requirements, knowing how to correctly apply them ensures that calculations are accurate and logical in any mathematical scenario.

Product Property of Logarithms

The product property of logarithms states that the logarithm of a product can be expressed as the sum of the logarithms of the individual factors, specifically:\[\log_b (MN) = \log_b M + \log_b N\]This property allows you to break down a logarithm of a combined product into separate parts. However, this property requires that both logarithms have the same base.
In the erroneous step of the solution, the expression \( \log_2 x + \log_3 y \) was mistakenly combined using this property. This is incorrect because the bases \(2\) and \(3\) differ, and thus, the property cannot be applied.

Quotient Property of Logarithms

The quotient property of logarithms is another essential tool for simplifying expressions. It states:\[\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N\]This property allows you to separate a division inside a logarithm into the subtraction of two logarithms.
Similar to the product property, it is vital to ensure that both terms have the same base before applying this property. In the solution provided, combining \(\log_6 xy - \log_4 z\) into a single term was mistakenly done without verifying equal bases. The incorrect use of this property contributed to the erroneous final expression \(\log_{24} xyz\).

Bases in Logarithms

The base of a logarithm is fundamentally important. It determines the context and the scaling of the output. In a proper application of logarithmic properties, all involved logarithms must share the same base.

• Base 10, also known as the common logarithm, is frequently used in scientific calculations.
• Base \(e\), the natural logarithm, is often used in calculus and exponential growth problems.
• Other bases, like \(2\), \(3\), or \(4\), might be used in specific problems, such as sequences and series or digital computations.

When attempting to simplify an expression such as \(\log_2 x + \log_3 y - \log_4 z\), the mistake made was neglecting the important rule that properties of logarithms, like the product and quotient properties, require identical bases to be applied. This oversight led to the incorrect attempt to simplify the expression across differing bases. Always check bases before manipulating logarithmic expressions.