Problem 36 - Precalculus

Question

Evaluate the logarithms exactly (if possible). $$\log _{5} 1$$

Step-by-Step Solution

Answer

The value of \( \log_{5} 1 \) is 0.

Step 1: Understanding the Logarithm Problem

We need to evaluate the expression \( \log_{5} 1 \). Recall that the logarithm \( \log_{b} a \) asks the question: "To what power must the base \( b \) be raised to obtain \( a \)?"

Step 2: Applying Logarithm Concepts

In \( \log_{5} 1 \), we are asking, "To what power must 5 be raised to equal 1?" It’s known that any number raised to the power of 0 is 1.

Step 3: Solving the Logarithm

Given the equation \( 5^x = 1 \), we realize that when \( x = 0 \), the equation holds true because \( 5^0 = 1 \). Hence, \( \log_{5} 1 = 0 \). This is the solution to the problem.

Key Concepts

Base of a Logarithm

When we talk about the base of a logarithm, it refers to the number that is raised to a certain power to obtain another number. Logarithms are written in the form of \( \log_b a \), where \( b \) is the base. This implies that we are seeking the exponent that makes \( b \) raised to this exponent equal \( a \).
For example, in \( \log_{5} 1 \), the base is 5. This means we are looking for a special exponent or power that, when 5 is raised to it, results in 1.

• Base influences how the logarithmic expression is evaluated.
• Changing the base changes the outcome of the logarithmic expression.
• A common base used in many logarithmic calculations is 10, known as the common logarithm, and the constant \( e \), known as the natural logarithm.

Understanding the concept of a base helps in comprehensively evaluating and solving logarithmic expressions.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions often involves determining the power to which a base number must be raised to result in another given number. This is the essence of logarithms.
To evaluate \( \log_{5} 1 \), we identify that it asks: "What power must 5 be raised to, to get 1?" We use the property of exponents stating that any number raised to the power of 0 results in 1.

• Apply the concept that \( b^0 = 1 \) for any base \( b \) not equal to zero. Therefore, \( 5^0 = 1 \).
• This indicates that \( \log_{5} 1 = 0 \), as 0 is the power we seek.
• Understanding this logic assists in evaluating similar logarithmic expressions.

Many logarithmic problems hinge on recognizing exponential relationships between the base and the evaluated number.

Exponents and Powers

Exponents and powers are closely related to logarithms, as they are essentially inverse operations. While exponents involve raising a base to a certain power to get a result, logarithms involve finding what that power is.
The fundamental idea is that if \( b^x = a \), then \( \log_b a = x \). Here, \( x \) is the exponent, and in the logarithmic form, it represents how many times we multiply the base by itself to reach \( a \).

• This relationship shows how logarithms can help reverse-engineer exponential equations.
• The property that \( b^0 = 1 \) is key when solving expressions like \( \log_{b} 1 \).
• For example, \( 5^0 = 1 \) confirms that the exponent needed to make the base equal to 1 is always zero, aiding in evaluating \( \log_{5} 1 \).

By understanding exponents and powers, students can more easily grasp logarithmic concepts, leading to more accurate evaluations of such expressions.