Problem 64 - Elementary and Intermediate Algebra

Question

Solve for \(x\). See Example 3 . $$ \log _{12} x=0 $$

Step-by-Step Solution

Answer

The solution is \( x = 1 \).

Step 1: Understanding the Logarithmic Equation

The given equation is \( \log_{12} x = 0 \). This states that the logarithm of \( x \) to the base 12 is 0. In logarithmic terms, this means that the power to which you must raise 12 to get \( x \) is 0.

Step 2: Convert Logarithmic to Exponential Form

According to the definition of logarithms, \( \log_{b}(a) = c \) means that \( b^c = a \). So, by converting the equation \( \log_{12} x = 0 \) into exponential form, we have \( 12^0 = x \).

Step 3: Calculate the Exponential Expression

Calculate \( 12^0 \). Any non-zero number raised to the power of 0 is equal to 1, so \( x = 12^0 = 1 \).

Key Concepts

Exponential Form

Converting a logarithmic equation to exponential form is a key step in solving logarithms. If you have an equation like \( \log_{b}(a) = c \), you are essentially saying that the base \( b \) raised to the power \( c \) equals \( a \). This transformation relies on the core relationship between exponential functions and logarithms, allowing us to move from one form to another easily.
For instance, in the given equation \( \log_{12} x = 0 \), converting it to exponential form involves raising the base 12 to the power of 0, i.e., \( 12^0 \). It is a straightforward conversion to say \( x = 12^0 \).
This step highlights the inverse nature of exponential and logarithmic functions, where moving from logarithmic to exponential form helps in finding unknowns such as \( x \) in equations. Whenever faced with a logarithmic equation, converting it to exponential form can be a powerful tool to simplify the problem.

Definition of Logarithms

The definition of a logarithm is fundamental in mathematics, particularly when solving equations involving logs. A logarithm, \( \log_{b}(a) = c \), essentially answers the question: "To which power must the base \( b \) be raised to produce the number \( a \)?"
This definition can help you to understand why \( \log_{b}(b^c) = c \). It’s saying that \( b \) raised to \( c \) gives you back \( b^c \). Therefore, if you see \( \log_{12} x \), and it's equal to a number like 0, it tells you that raising the base 12 to the power of 0 must result in the original number \( x \).
Through grasping the concept of what a logarithm is, you recognize its role as the inverse operation of exponentiation. Understanding this relationship allows you to tackle more complex logarithmic problems with clarity and ease.

Logarithm Properties

Logarithms come with a set of useful properties that can simplify calculations and aid in solving equations. Here are some key properties:

• Product Property: \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \)
• Quotient Property: \( \log_{b}\left(\frac{m}{n}\right) = \log_{b}(m) - \log_{b}(n) \)
• Power Property: \( \log_{b}(m^n) = n \cdot \log_{b}(m) \)
• Change of Base Formula: \( \log_{b}(a) = \frac{\log_{k}(a)}{\log_{k}(b)} \) for any base \( k \).

These properties are based on the logarithms’ ability to convert multiplication into addition, division into subtraction, and powers into products. They form the foundation of many logarithmic manipulations and problem-solving techniques.
To solve the equation \( \log_{12} x = 0 \), you might not need these properties directly, but understanding them can provide a deeper insight into the behavior and rules governing logarithms. In more complex problems, applying these properties can simplify and resolve otherwise convoluted expressions.