Problem 74 - Intermediate Algebra

Question

Find each value of \(x .\) $$ \log _{12} x=0 $$

Step-by-Step Solution

Answer

The value of \( x \) is 1.

Step 1: Understand the Logarithm Definition

The equation \( \log_{12} x = 0 \) asks us to determine the value of \( x \) for which the logarithm to the base 12 is zero.

Step 2: Convert to Exponential Form

Using the definition of a logarithm, we convert the equation \( \log_{12} x = 0 \) to its exponential form: \( x = 12^0 \).

Step 3: Simplify the Exponential Expression

Calculating \( 12^0 \), we know that any non-zero number raised to the power of 0 is 1. So we have \( x = 1 \).

Key Concepts

Exponential Form

When dealing with logarithms, it's often helpful to convert them into an exponential form to better understand their meaning and solve equations. In essence, the exponential form is a way of expressing numbers using a specific base and an exponent. This form is expressed as \( a^b \), where \( a \) is the base and \( b \) is the exponent. To illustrate this with an example, consider the logarithmic equation \( \log_{12} x = 0 \). To convert this into its exponential form, we interpret it as: "What exponent \( b \) do I raise the base \( 12 \) to, to get \( x \)?" Thus, the exponential form is \( x = 12^0 \). By changing the logarithmic equation into the exponential form, we can make direct calculations, simplifying the problem-solving process.

Logarithm Definition

Logarithms are often seen as the inverses of exponents. Understanding this inverse relationship is crucial when solving logarithmic problems. A logarithm answers the question: "To what power must we raise a base \( b \) to obtain a number \( x \)?"This is formally expressed in the formula \( \log_b x = y \), which is equivalent to \( b^y = x \). In layman terms, if you have a number \( x \) and you want to express it in terms of an exponential base \( b \), the logarithm with base \( b \) gives you the exponent \( y \).For example, \( \log_{12} x = 0 \) means that the base 12 raised to the power of 0 equals \( x \). Logarithms can initially seem complex, but they are just another way to express relationships in numbers, making them powerful in various areas of mathematics.

Power of Zero

One of the fundamental rules of exponents is the 'power of zero' rule. This rule states that any non-zero base raised to the power of zero equals one. It means \( a^0 = 1 \) for any non-zero number \( a \).Understanding this rule is crucial when simplifying equations involving exponents. For instance, in our exercise, converting \( \log_{12} x = 0 \) to exponential form gives \( x = 12^0 \). According to the power of zero rule, \( 12^0 = 1 \), which simplifies our solution and allows us to solve for \( x \). This rule exemplifies the importance of foundational laws in mathematics, as simple principles like 'power of zero' help unravel more complicated equations smoothly.