Problem 59 - Algebra 2

Question

Write true or false for each statement. Justify your answer. \(\log _{3} \frac{3}{2}=\frac{1}{2} \log _{3} 3\)

Step-by-Step Solution

Answer

The statement is false.

Step 1: Simplification

Firstly, simplify the right hand side of the equation. According to the logarithmic identity \(log_b(b)=1\), \(log_3(3)=1\). Therefore, \(\frac{1}{2} \cdot log_3(3) = \frac{1}{2} \cdot 1 = 0.5\)

Step 2: Evaluate

Now, evaluate the left-hand side. Since 3 raised to the power 0.5 (or 1/2) does not equal 3/2, therefore \(log_3 \frac{3}{2}\) is not equal to 0.5 as it doesn't fulfill the property of logarithms where \(log_b(b^m) = m \)

Key Concepts

Logarithmic Identities

Logarithmic identities are useful mathematical tools to simplify and solve logarithmic expressions. One of the most fundamental identities is that the logarithm of a number to its own base is 1; for example, \( \log_b(b) = 1 \). This identity works because any number raised to the power of 1 is itself.

In the given exercise, this identity helps simplify the expression \( \log_3(3) \) to 1, making calculations much easier. Another important identity is the power rule: \( \log_b(x^m) = m \cdot \log_b(x) \). This allows us to manipulate logarithms involving exponents, breaking them into manageable parts.

• **Change of Base Formula**: This is used to convert a logarithm of one base to another: \( \log_a(x) = \frac{\log_b(x)}{\log_b(a)} \).
• **Addition and Subtraction Rules**: \( \log_b(xy) = \log_b(x) + \log_b(y) \) and \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \).

These identities ensure that we can manipulate logarithmic expressions confidently, making problem-solving more manageable.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. When dealing with logarithms, understanding exponents is crucial because logarithms are the inverse operation of exponentiation. This means that if \(b^y = x\), then \(\log_b(x) = y\).

In the context of the original exercise, this concept helps us see that if \(3^{0.5} eq \frac{3}{2}\), then \(\log_3\left(\frac{3}{2}\right)\) does not equal \(0.5\).

• **Multiplication**: \(a^m \times a^n = a^{m+n}\).
• **Division**: \(a^m / a^n = a^{m-n}\).
• **Power of a Power**: \((a^m)^n = a^{m \cdot n}\).

Understanding exponents allows us to grasp why logarithms work the way they do, aiding in the simplification and solving of logarithmic equations.

Logarithmic Equations

Logarithmic equations involve solving for unknown variables by applying logarithmic identities and properties. These equations can often seem complex, but breaking them down step-by-step using logarithmic rules makes them approachable.

In the exercise provided, the task is to verify whether two logarithmic expressions are equal. Since evaluating \(\log_3\left(\frac{3}{2}\right)\) and comparing it to the simplified \(0.5\) through established rules didn’t hold true, the equation was false. Here are some steps for dealing with logarithmic equations:

• **Isolate the Logarithm**: Like any algebraic equation, try to get the log expression alone on one side.
• **Convert to Exponential Form**: Swap the log for its exponential equivalent, which can make solving easier.
• **Use Identities**: Apply rules like product, quotient, and power to simplify.

Understanding how to manipulate and solve logarithmic equations is essential in algebra and helps solve real-world problems involving exponential growth or decay.