Problem 67 - Algebra 2

Question

Write true or false for each statement. Justify your answer. \(\log _{2} 3+\log _{3} 2=\log _{6} 6\)

Step-by-Step Solution

Answer

The statement '\(\log _{2} 3+\log _{3} 2=\log _{6} 6\)' is False

Step 1: Simplify Left Side

First simplify the left side of the equation. By applying the change of base formula, we convert the bases of both the logarithms to a common base. Let's choose base 10. The formula is \(\log_a b = \frac{\log_c b}{\log_c a}\). We apply this formula as follows: \(\log _{2} 3+\log _{3} 2= \frac{\log _{10} 3}{\log _{10} 2} + \frac{\log _{10} 2}{\log _{10} 3}\)

Step 2: Simplify Right Side

The right side of the equation is \(\log _{6} 6\). Using the property of log that \(\log_b b = 1\), this simplifies to 1.

Step 3: Compare Both Sides

By comparing steps 1 and 2, it becomes clear that left side is not equal to the right side. Therefore, the equation \(\log _{2} 3+\log _{3} 2=\log _{6} 6\) is not true.

Key Concepts

Change of Base Formula

The change of base formula is an essential tool when dealing with logarithms of different bases. It's like a translator that allows us to shift a logarithm into a base we're more comfortable with. The general formula is \(\log_a b = \frac{\log_c b}{\log_c a}\),which allows us to turn any logarithm with base \(a\) into one with base \(c\).
Let's say you have \(\log_2 3\) and you want to convert it to base 10 (a more common base in calculations). You'll use the formula like this:

• First, calculate \(\log_{10} 3\)
• Then, calculate \(\log_{10} 2\)
• Finally, divide the two results as per the formula: \(\frac{\log_{10} 3}{\log_{10} 2}\)

This method is quite handy when you're solving equations that have logarithms with different bases, as it allows you to work with a single base consistently throughout your calculations.

Logarithm Simplification

Simplifying logarithmic expressions can make solving equations much more manageable. One crucial property is that \(\log_b b = 1\) because any number raised to the power of 1 is the number itself. So, for example, \(\log_6 6 = 1\).
Another simplification trick involves combining terms. If you have two logarithms you want to add, such as \(\log_a x + \log_a y\), you can combine them into a single term using multiplication: \(\log_a(xy)\). But in our original problem, we had logarithms of different bases, which required using the change of base formula instead.
Whenever you face multiple logarithms, close attention should be paid to any potential simplifications to reduce complexity, achieving clarity and more straightforward solutions.

Logarithm Equality Verification

Verifying whether two logarithmic expressions equal is an important step in solving logarithmic equations. This often involves comparing the simplified forms of each expression.
For instance, if you took our original equation \(\log_2 3 + \log_3 2 = \log_6 6\), you would first simplify both sides: the left by converting both terms to a common base, and the right by recognizing it equals 1 due to the property \(\log_b b = 1\). Next, you compare these simplifications.
In this case, after comparing \(\frac{\log_{10} 3}{\log_{10} 2} + \frac{\log_{10} 2}{\log_{10} 3}\) with 1, you'll see they don't match, thus verifying the equation's inequality. This is a fundamental part of solving logarithmic equations, ensuring that any assumptions or transformations led to correct results.