Problem 58 - Elementary and Intermediate Algebra

Question

Solve for \(x\). See Example 3 . $$ \log _{3} x=-4 $$

Step-by-Step Solution

Answer

\(x = \frac{1}{81}\).

Step 1: Understand the Logarithm Equation

The given equation is \(\log_{3} x = -4\). This equation asks which power of 3 gives \(x\).

Step 2: Convert Logarithmic Form to Exponential Form

To solve the equation \(\log_{3} x = -4\), convert it into its exponential form using the property: \(\log_{b} a = c\) implies \(b^c = a\). Here, \(b = 3\), \(a = x\), and \(c = -4\). Thus, the equation becomes \(3^{-4} = x\).

Step 3: Calculate the Exponential Expression

Now calculate \(3^{-4}\). This means taking the reciprocal of 3 raised to the power of 4, as negative exponent indicates reciprocal: \(3^{-4} = \frac{1}{3^4}\).

Step 4: Compute Powers

Compute \(3^4\), which is \(3 \times 3 \times 3 \times 3 = 81\). Thus, \(3^{-4} = \frac{1}{81}\).

Step 5: Write the Solution

From the previous steps, we found that \(x = \frac{1}{81}\). This is the solution to the equation \(\log_{3} x = -4\).

Key Concepts

Exponential Form

The concept of exponential form is a cornerstone in understanding logarithms. When we express a logarithmic equation in exponential form, we uncover the "secret" life of the equation by showing its power base structure.

In simpler terms:

• A logarithm like \( \log_b a = c \) translates to \( b^c = a \). This means the base \( b \) raised to the power of \( c \) equals \( a \).
• It's turning a logarithm back into its exponential "story".

Adopting this viewpoint plays a crucial role in solving logarithmic equations because it helps us identify the relationship between the components. This element was pivotal in the example, transforming \( \log_{3} x = -4 \) into the exponential equation \( 3^{-4} = x \).

Thus, the process of converting a logarithmic equation to an exponential form is like shifting perspectives from "what power of this number...?” to "this number raised to this power gives...".

Negative Exponents

Negative exponents might seem a bit mysterious at first, but they are quite straightforward once you familiarize yourself with them. They provide a clever way to express reciprocals.

Here's how they work:

• A negative exponent like \( b^{-n} \) means \( \frac{1}{b^n} \).
• It's like flipping the base to the denominator and then raising it to the positive exponent.

In the context of our example, \( 3^{-4} \) was translated as \( \frac{1}{3^4} \). This transformation is essential because it connects the dots between exponents and fractions, allowing us to solve for \( x \) by correctly evaluating \( 3^{-4} \).

Understanding negative exponents is vital because they frequently appear in equations and functions. Mastering them unlocks the ability to solve not just logarithmic, but also many types of algebraic problems.

Powers of Numbers

Powers of numbers is a fundamental concept in both algebra and the broader scope of mathematics. Powers are used to express repeated multiplication succinctly.

Let's break it down:

• When you see \( b^n \), it means multiplying the base \( b \) by itself \( n \) times.
• In our equation, \( 3^4 \) is calculated as \( 3 \times 3 \times 3 \times 3 \), which results in 81.

Understanding how to evaluate powers helps in grasping the magnitude and scale of numbers.

It's useful in contexts ranging from simple arithmetic to complex scientific calculations. In solutions like the one provided, accurately calculating the power \( 3^4 \) was key in determining the exact value of \( x = \frac{1}{81} \).

With this skill, you can tackle various mathematical challenges, providing a solid base for further studies in mathematics.