Problem 89 - Precalculus

Question

Determine whether each statement is true or false. The sum of logarithms with the same base is equal to the logarithm of the product.

Step-by-Step Solution

Answer

True, the statement is correct as verified by the property calculation.

Step 1: Understand the Logarithm Property

The property that the sum of logarithms with the same base is equal to the logarithm of the product can be expressed as: \[\log_b(x) + \log_b(y) = \log_b(xy)\] Here, \(b\) is the base of the logarithms, \(x\) and \(y\) are the values for which the logarithm is calculated.

Step 2: Apply the Property to Verify

To check if the property holds, consider an example. Suppose we have \(\log_2(3) + \log_2(4)\). According to the property:\[\log_2(3) + \log_2(4) = \log_2(3 \times 4) = \log_2(12)\]Check if this equation holds by considering the values of these expressions.

Step 3: Compare Calculated Logarithms

First, calculate \(\log_2(3) + \log_2(4)\) using the property:\[\log_2(3) + \log_2(4) = \log_2(12)\]Next, find \(\log_2(12)\):\(\log_2(12)\) evaluates to approximately 3.585, which should be the sum of \(\log_2(3)\) and \(\log_2(4)\) when calculated separately.

Step 4: Conclusion

The calculations confirm that the sum of logarithms \(\log_2(3) + \log_2(4)\) results in the same value as \(\log_2(12)\). This demonstrates that the sum of logarithms with the same base equals the logarithm of the product, and hence, the statement is true.

Key Concepts

Logarithm Addition Rule

The logarithm addition rule is an important property that simplifies calculations involving logarithms. It states that the sum of two logarithms with the same base can be combined into a single logarithm with the product of the original arguments.

• Formula: \( \log_b(x) + \log_b(y) = \log_b(xy) \)

Here, the base \( b \) must remain consistent across both terms. To see this in action, consider the expression \( \log_2(3) + \log_2(4) \). Using the addition rule, this simplifies to \( \log_2(12) \).
The intuitive idea behind this rule is that multiplying numbers together increases their combined logarithmic scale. This aids in simplifications during algebraic manipulations. Understanding this rule not only saves time but also unveils deeper relationships in logarithmic expressions.

Logarithmic Equations

Logarithmic equations are equations that involve logarithms of unknown variables. Solving these requires an understanding of logarithm properties like the addition and change of base rules.
A typical structure of a logarithmic equation might appear as \( \log_b(x) + \log_b(y) = c \), where \( c \) is a constant. To solve it, you apply the logarithm addition rule. This yields \( \log_b(xy) = c \), where you then convert the logarithmic equation into its exponential form: \( xy = b^c \).

• Key Steps in Solving:
• Apply logarithm properties to simplify.
• Convert the log equation to its exponential form.
• Solve the resulting exponential equation for \( x \) or \( y \).

Working with logarithmic equations often requires checking if solutions solve the original equation, as not all calculations lead to valid log values. Remember, the arguments inside a logarithm must always be positive.

Logarithm Base Conversion

In many problems, you may need to convert logarithms of one base to another. This is essential in computations where calculators are involved or different logarithm bases are used.
The base conversion formula is: \[\log_b(x) = \frac{\log_k(x)}{\log_k(b)}\]Here, \( b \) is the original base and \( k \) is the new base (commonly base 10 or base \( e \) for natural logs).

• Understanding Base Conversion:
• Use when needing a specific logarithmic base for calculations.
• This allows flexibility in handling logarithms with any base.
• Essential for solving problems not easily computed in the original base.

One practical example is converting natural logarithms to base 10, which some calculators or contexts prefer. With this knowledge, solving logarithms becomes more versatile and flexible irrespective of the base used initially.