Problem 16 - Elementary and Intermediate Algebra

Question

Does \(\frac{\log 7}{\log 3}=\log 7-\log 3 ?\)

Step-by-Step Solution

Answer

No, \(\frac{\log 7}{\log 3}\) is not equal to \(\log 7 - \log 3\).

Step 1: Understanding the Given Equation

We are given the equation \(\frac{\log 7}{\log 3} = \log 7 - \log 3\) and are asked to check if it is true. We need to manipulate each side of the equation separately to see if they are equal.

Step 2: Analyzing the Left Side

The left hand side of the equation is \(\frac{\log 7}{\log 3}\). This represents the logarithm of 7 with base 3, which is written as \(\log_3 7\).

Step 3: Analyzing the Right Side

The right hand side of the equation is \(\log 7 - \log 3\). By the properties of logarithms, this can be rewritten as \(\log\left(\frac{7}{3}\right)\).

Step 4: Comparing Both Sides

We need to check if \(\log_3 7 = \log\left(\frac{7}{3}\right)\). In general, \(\log_b a eq \log(a/b)\) for most values of \(a\) and \(b\). Therefore, we conclude that the two expressions are not equal.

Key Concepts

Properties of Logarithms

Logarithms have several useful properties that can make complex expressions easier to handle. One of these properties is that the logarithm of a quotient is the difference of the logarithms, expressed as:

• \( \log_b (\frac{a}{c}) = \log_b a - \log_b c \).

This is what we see in the original step where the right-hand side \( \log 7 - \log 3 \) was transformed into \( \log(\frac{7}{3}) \). Understanding this property allows us to rewrite and solve logarithmic expressions and equations efficiently.
Another important property is the change of base formula, which we'll discuss next. This allows us to compute logarithms in terms of different bases and is critical for using calculators effectively.

Logarithmic Equations

Logarithmic equations involve logarithms and typically require the manipulation of their properties to solve. When faced with such equations, like the one in the exercise, comparing both sides is crucial.
In our example, we were given:

• \( \frac{\log 7}{\log 3} = \log 7 - \log 3 \).

To determine the validity, we applied the properties of logarithms. By rewriting \( \log 7 - \log 3 \) as \( \log(\frac{7}{3}) \), we could directly compare it with \( \log_3 7 \). Recognizing when an equation's sides are not equal is often crucial. Despite the temptation to consider them equivalent, a deeper understanding shows the difference.

Change of Base Formula

The change of base formula is a powerful tool in computing logarithms with different bases. It allows us to express logs in terms of any base we're comfortable with, typically base 10 or base \( e \).
The formula itself is:

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

This was indirectly used in the solution by recognizing that \( \frac{\log 7}{\log 3} \) is equivalent to \( \log_3 7 \).
Understanding and applying this formula is essential when working beyond simple logarithmic problems. It allows for calculations that are otherwise complicated, making it an invaluable asset in solving equations and understanding their relationship to different bases.