Problem 23 - Algebra A Combined Function

Question

Find the exact value of each logarithm. $$ \log 10^{3} $$

Step-by-Step Solution

Answer

The exact value of \( \log 10^3 \) is 3.

Step 1: Understand the Logarithm Concept

The logarithm \( \log_b(x) \) asks the question: "To what power must the base \( b \) be raised, to obtain \( x \)?" Here, the logarithm is \( \log (10^{3}) \), which means how many times must 10 be multiplied by itself to get \( 10^3 \).

Step 2: Apply the Power Rule of Logarithms

According to the power rule of logarithms, \( \log_b(a^n) = n \cdot \log_b(a) \). Therefore, \( \log (10^3) = 3 \cdot \log (10) \).

Step 3: Evaluate \( \log (10) \)

We know from properties of logarithms that \( \log_{10}(10) = 1 \) because 10 raised to the power of 1 is 10. Therefore, \( \log (10) = 1 \).

Step 4: Calculate the Final Value

Substitute the value from Step 3 back into the equation from Step 2: \( \log (10^3) = 3 \cdot 1 = 3 \). Thus, the exact value of the logarithm is 3.

Key Concepts

Power Rule of Logarithms

When dealing with logarithms involving exponents, the power rule is incredibly useful. It simplifies expressions by moving the exponent in front of the logarithm, thus making calculations more straightforward. The power rule states that for any positive number \(a\), base \(b\), and exponent \(n\), \( \log_b(a^n) = n \cdot \log_b(a) \). The logic behind this is based on the properties of exponents themselves.For example, in the exercise \( \log (10^3) \), we see \(3\) as the exponent of \(10\). By applying the power rule, we move \(3\) in front, obtaining \(3 \cdot \log (10)\). This transformation is key because it allows us to break down complex expressions into more manageable parts, making logarithmic calculations significantly easier and more intuitive.

Properties of Logarithms

Logarithms have several properties that make them versatile tools for mathematical operations. Understanding these properties helps us manipulate logarithmic expressions efficiently. Key properties include:

• Product Rule: \( \log_b(xy) = \log_b(x) + \log_b(y) \)
• Quotient Rule: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \)
• Power Rule: \( \log_b(x^n) = n \cdot \log_b(x) \)
• Logarithm of 1: \( \log_b(1) = 0 \) for any base \( b \)
• Base Rule: For any number that is the same as the base of the logarithm, \( \log_b(b) = 1 \)

In the exercise, we particularly used the base rule where \( \log_{10}(10) = 1 \). Recognizing and employing these properties can greatly simplify the evaluation process, enabling us to handle logarithmic expressions with confidence.

Base of Logarithm

The base of a logarithm is a fundamental component that determines how we interpret the logarithm. It is the number that we raise to a specific power to reach another number. Typically, the base is written as a subscript in the logarithmic expression \( \log_b(x) \). In the context of the exercise, we implicitly deal with base 10, often referred to as the common logarithm. Common logarithms simplify our calculation since converting numbers back and forth involving the base 10 is intuitive. We know that \( \log_{10}(10) = 1 \) because 10 raised to the power of 1 equals 10. This base is particularly useful in many scientific and practical applications due to its alignment with the decimal system, which is widely used in everyday calculations. Different bases like \(2\) or \(e\) can be used in specific contexts, but understanding base 10 gives a solid foundation for further exploration in the realm of logarithms.