Problem 28 - College Algebra

Question

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{5} x=3$$

Step-by-Step Solution

Answer

The solution to the equation \( \log _{5} x = 3 \) is \( x = 125 \).

Step 1 : Convert to Exponential Form

The logarithmic equation \( \log _{5} x = 3 \) can be written in exponential form as \( 5^3 = x \).

Step 2 : Simplify the Equation

Calculating \( 5^3 \) results in \( 125 = x \).

Step 3 : Verify the Solution

Check that \( x = 125 \) does not produce the logarithm of a negative number or zero. The argument of the logarithm is \( x = 125 \), which is greater than zero.

Key Concepts

Solving Logarithmic Equations

Solving logarithmic equations can seem daunting at first, but with some simple tools and understanding, it becomes straightforward. Logarithmic equations often require us to unwind the logarithm to find the value of the variable.

Here is how you can approach solving them:

• Firstly, understand the basic structure of a logarithm. The equation is generally written in the form \( \log_b a = c \), meaning the logarithm of \( a \) to base \( b \) equals \( c \).
• The goal is to isolate the variable, typically within the logarithmic function.
• Convert the logarithm into its equivalent exponential form to solve for the variable easily.

Approaching logarithmic equations in a systematic way ensures that you correctly handle all parts of the equation without errors. The key is to ensure all operations maintain the validity of the equation and any transformations abide by mathematical rules.

Exponential Form

Understanding how to convert a logarithmic equation into exponential form is crucial for solving it. A logarithmic equation \( \log_b x = c \) can be rewritten in exponential form as \( b^c = x \).

This process involves taking the base \( b \) of the logarithm and raising it to the power \( c \), which is the number the logarithm equals. This step effectively "undoes" the logarithm, leaving you with the variable \( x \) isolated on one side of the equation.

For example, consider the equation \( \log_{5} x = 3 \). Converting it to exponential form gives \( 5^3 = x \). Here, base 5 is raised to the power of 3, solving the equation for \( x \). Applications of this conversion are numerous and allow you to solve the equation in a manageable form. Always double-check your work by back-substituting the solution into the original equation to ensure it holds true.

Logarithmic Properties

Understanding the properties of logarithms is fundamental to solving logarithmic equations. These properties allow you to manipulate logarithms to simplify and solve equations. Some key logarithmic properties include:

• The Product Rule: \( \log_b(m \, n) = \log_b m + \log_b n \)
• The Quotient Rule: \( \log_b(m/n) = \log_b m - \log_b n \)
• The Power Rule: \( \log_b(m^n) = n \, \log_b m \)

These properties apply universally, providing powerful tools for breaking down complex logarithmic expressions into simpler components.

For instance, knowing these properties can help identify opportunities to factor or expand other terms within the equation. Consistent familiarity with these rules enhances your flexibility to tackle a wide range of logarithmic problems, beyond just simple equations.

Base of Logarithm

The base of a logarithm is a pivotal component and determines the operations conducted on the logarithmic expression. In equations like \( \log_b a = c \), the base \( b \) tells you what number you are repeatedly multiplying to get \( a \).

The base is written as a subscript in logarithmic equations and is fundamental when converting to exponential form, where \( a = b^c \).

Choosing the correct base can simplify the whole problem-solving process by directly impacting how you interpret and manipulate the logarithmic expression. For example, in the equation \( \log_5 x = 3 \), the base is 5. This means multiplying 5 three times in exponential form gives you \( x = 5^3 = 125 \).

Understanding the base and its role helps prevent mistakes in calculations and interpretations, offering clearer pathways to solutions in logarithmic equations.