Problem 48 - Elementary and Intermediate Algebra

Question

Solve each equation. See Example \(6 .\) $$ \log \frac{1}{5} x=-3 $$

Step-by-Step Solution

Answer

The solution is \( x = 0.005 \).

Step 1: Understand the Equation

We are given the logarithmic equation \( \log \frac{1}{5}x = -3 \). This means the logarithm of the expression \( \frac{1}{5}x \) is equal to \(-3\).

Step 2: Convert Logarithmic Equation to Exponential Form

The expression \( \log \frac{1}{5}x = -3 \) means \( \frac{1}{5}x = 10^{-3} \), by the definition of a logarithm: \( \log_b a = c \) means \( a = b^c \).

Step 3: Solve for \( x \)

Now, we solve the exponential equation \( \frac{1}{5}x = 10^{-3} \). Multiply both sides by 5 to isolate \( x \): \[ x = 5 \times 10^{-3} \] Calculate the right-hand side: \[ x = 0.005 \]

Step 4: Verify the Solution

Substitute \( x = 0.005 \) back into the original equation to check: \[ \log \frac{1}{5}(0.005) = \log 0.001 = -3 \]. The solution satisfies the original equation, confirming that \( x = 0.005 \) is correct.

Key Concepts

Exponential Equations

Exponential equations are equations where variables appear in exponents. They're foundational in mathematics because they depict constant growth or decay patterns, often observed in nature and finance.

To understand these equations, consider the general form:

• General Form: If you have an equation in the form of \( a = b^x \), where \( b \) is a constant base and \( x \) is an exponent, then you're dealing with an exponential equation.
• Utility: Exponential equations are used to model situations like population growth, radioactive decay, and compound interest, where something grows or shrinks at a rate proportional to its current value.
• Solving Tips: Transforming an exponential equation into a simpler form can often involve using logarithms to "bring down" the exponent. This is key in making the variable accessible for solving.

Understanding how to manipulate these equations is crucial, especially when dealing with real-world problems. The transformation into logarithmic form is an elegant way to make them more manageable.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They are crucial in algebra because they allow you to solve complex equations by converting them to a simpler form.

Here's a deeper dive into the essence of logarithmic functions:

• Definition: The logarithm of a number is the exponent to which the base must be raised to produce that number. For example, in the equation \( \,\log_b a = c \, \), it translates to \( \, b^c = a \, \).
• Properties: There are vital properties of logarithms that make them useful, such as the product rule, the quotient rule, and the power rule. These properties help in manipulating logarithmic expressions effectively.
• Applications: Logarithms are widely used in scientific calculations, such as measuring the intensity of earthquakes on the Richter scale, sound intensity in decibels, and in fields like finance and information theory.

In essence, logarithms simplify complex equations by converting exponential relationships into linear ones, making them easier to solve.

Solving Equations

Solving equations involves finding the variable value that makes the equation true. It's a core concept in mathematics, as it allows us to solve real-world problems by modeling them with mathematical expressions.

Here’s how to efficiently approach solving logarithmic and exponential equations:

• Identify the Type: Recognize if you're dealing with exponential or logarithmic equations. This helps in choosing the appropriate method to solve them.
• Convert and Simplify: Logarithmic equations can often be transformed into exponential equations. This conversion uses the property \( \log_b a = c \Rightarrow a = b^c \), greatly simplifying the process.
• Check Your Work: Always substitute your solution back into the original equation to verify its correctness. This step ensures that no computational errors were made and the solution is accurate.
• Practical Application: Whether you're analyzing population trends, or calculating financial growth, being proficient in solving equations expands your analytical toolkit.

Solving equations allows us to translate theoretical math into meaningful interpretations, whether it's in pure mathematics or applied fields.