Problem 55 - Elementary and Intermediate Algebra

Question

Solve for \(x\). See Example 3 . $$ \log _{5} 125=x $$

Step-by-Step Solution

Answer

The value of \(x\) is 3.

Step 1: Understand the Logarithmic Equation

The given equation is \( \log_{5} 125 = x \). This is a logarithmic equation where 5 is the base, 125 is the number, and \(x\) is the exponent. Our task is to find the value of \(x\) such that \(5^x = 125\).

Step 2: Convert to Exponential Form

Rewrite the logarithmic equation \( \log_{5} 125 = x \) as an exponential equation using the property: if \( \log_b a = x \), then \( b^x = a \). Thus, \( 5^x = 125 \).

Step 3: Express 125 as a Power of 5

To solve \( 5^x = 125 \), express 125 as a power of 5. Notice that 125 can be written as \( 5^3 \) because \( 5 \times 5 \times 5 = 125 \).

Step 4: Equalize the Exponents

Since \( 5^x = 5^3 \), and the bases are the same, we can set the exponents equal to one another: \( x = 3 \).

Step 5: Verify the Solution

Verify our solution by substituting \( x = 3 \) back into the original equation. Compute \( 5^3 \) to confirm it equals 125. Since \( 5^3 = 125 \), our solution \( x = 3 \) is correct.

Key Concepts

Exponential Equations

Exponential equations are mathematical statements where variables appear as exponents. These equations generally take the form \( b^x = a \), where \( b \) is the base, \( x \) is the exponent or variable we need to solve for, and \( a \) is the result. Understanding exponential equations is key to solving many problems in algebra since they often appear in natural growth models, financial calculations, and scientific computations.

Here are some quick insights into solving exponential equations:

• Make sure that you express both sides of the equation with the same base, if possible.
• If the bases are equal, you can set the exponents equal to each other and solve for the unknown.
• Use logarithms when the bases cannot be easily made to be the same.

Mastering exponential equations helps build the groundwork needed to understand more complex mathematical topics, making them a fundamental component of higher-level math learning.

Logarithmic Properties

Logarithmic properties are essential tools for manipulating and solving equations involving logarithms. Understanding these properties allows you to convert between exponential and logarithmic forms easily.

Here are some core logarithmic properties:

• **Product Rule:** \( \log_b (mn) = \log_b m + \log_b n \), which allows splitting the log of a product into a sum of logs.
• **Quotient Rule:** \( \log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n \), useful for separating the log of a division into a difference.
• **Power Rule:** \( \log_b (m^n) = n \cdot \log_b m \), which puts exponents out in front as a factor of the log.
• **Change of Base Formula:** \( \log_b a = \frac{\log_c a}{\log_c b} \), crucial for calculating logs in any base using a calculator.

These properties allow you to simplify complex logarithmic expressions and solve equations efficiently by converting them to a format that's easier to interpret.

Solving Logarithmic Expressions

When solving logarithmic expressions, the goal is usually to find the value of the variable by isolating the log term before applying exponential principles. It's important to switch between the logarithmic and exponential forms as needed to uncover what the unknown represents. Solving these expressions often involves combining both logarithmic and exponential knowledge.

Here's a general guideline for solving logarithmic equations:

• Look to isolate the logarithm on one side whenever possible.
• Use the inverse operation of the logarithm, which is exponentiation, to remove it.
• If dealing with multiple logs, employ logarithmic properties to consolidate them.
• Finally, solve the resulting exponential equation using methods for exponential equations.

Whether you're converting \( \log_{5} 125 = x \) to an exponential form like \( 5^x = 125 \), or simplifying complex logs into sums and differences, these strategies work hand-in-hand to find solutions with ease.