Problem 48 - Intermediate Algebra

Question

Solve each equation. $$ \log \frac{1}{5} x=-3 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{1}{200} \).

Step 1: Understand the Logarithmic Equation

The given equation is in the form of a logarithm: \( \log_{10} \frac{1}{5} x = -3 \). This means that \( 10^{-3} = \frac{1}{5} x \). We need to solve for \( x \).

Step 2: Convert the Logarithmic Equation to an Exponential Equation

Since \( \log_{10} \frac{1}{5} x = -3 \) implies \( 10^{-3} = \frac{1}{5} x \), rewrite the equation as \( \frac{1}{5} x = \frac{1}{1000} \).

Step 3: Solve for \( x \)

To isolate \( x \), multiply both sides of the equation \( \frac{1}{5} x = \frac{1}{1000} \) by 5. This gives us \( x = \frac{5}{1000} = \frac{1}{200} \).

Step 4: Verify the Solution

Substitute \( x = \frac{1}{200} \) back into the original logarithmic equation \( \log_{10} \frac{1}{5} x = -3 \). Calculate \( \frac{1}{5} \times \frac{1}{200} = \frac{1}{1000} \) and \( \log_{10} \frac{1}{1000} = -3 \). This confirms that the solution is correct.

Key Concepts

Exponential Equations

An exponential equation is an equation in which a variable appears in the exponent. This is interesting because, unlike linear or quadratic equations, exponential equations can grow very quickly. In the context of solving logarithmic equations, you often convert the logarithmic form to an exponential form to simplify the process. For example, if you have a logarithmic equation like \( \log_{10} x = y \), it implies that x is equal to 10 raised to the power of y.
This method of conversion is helpful because it leverages our knowledge of exponential functions, which are more straightforward to solve. In the given exercise, the logarithmic equation was converted to its exponential form: \( 10^{-3} = \frac{1}{5} x \). By rewriting the equation this way, we simplify the task of finding the unknown variable, in this case, the variable \( x \).

Solving for x

Solving for \( x \) is the key action in many algebra problems, including equations with logarithms and exponents. Once you have manipulated the equation to a simpler form—through methods such as converting a logarithmic equation into an exponential one—you can focus on isolating the variable.
For instance, in the step \( \frac{1}{5} x = \frac{1}{1000} \), you want to get \( x \) by itself on one side of the equation. To do this, multiply both sides of the equation by 5, which leaves you with \( x = \frac{5}{1000} \).
It's essential to keep operations balanced—what you do to one side, you must do to the other—to accurately solve any equation. The solution \( x = \frac{5}{1000} \) simplifies further to \( x = \frac{1}{200} \), giving you a precise value for \( x \).

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions, and understanding them is crucial for solving equations involving logs. The notation \( \log_{b} a = c \) means that \( b^c = a \). Logs tell you what power you need to raise the base to, in order to get the number a.
For the problem \( \log \frac{1}{5} x = -3 \), it implies that you need to find the number \( x \) that when multiplied by \( \frac{1}{5} \), gives you \( 10^{-3} \). This ability to relate multiplication and division directly to powers or exponents is why logs are so useful in solving complex equations.
They help you break down problems into smaller, manageable parts, making it easier to find a solution. Understanding the role of logarithms provides you with the toolset to tackle not just textbook exercises but other real-world applications where growth rates and exponential changes occur.