Problem 12 - College Algebra

Question

Write each equation in its equivalent logarithmic form. $$ 5^{-3}=\frac{1}{125} $$

Step-by-Step Solution

Answer

The equivalent logarithmic form of the equation \(5^{-3} = \frac{1}{125}\) is \( \log_{5}{(\frac{1}{125})} = -3 \).

Step 1: Identify the Base, Exponent and Result

From the equation \(5^{-3} = \frac{1}{125}\), we can see that the base (b) is 5, the exponent (y) is -3 and the result (x) is \( \frac{1}{125} \).

Step 2: Convert to Logarithmic Form

We substitute the base, exponent and the result into logarithmic format \(\log_b{x} = y\), which gives the equivalent logarithmic form as \( \log_{5}{(\frac{1}{125})} = -3 \).

Key Concepts

Base and Exponent

When we talk about expressions like \(5^{-3} = \frac{1}{125}\), we are looking at a base and an exponent working together to create a number. Let's break these terms down a bit more.

- **Base**: In a power expression, the base is the number that is being multiplied. It's the starting point, if you will. In our example, the base is \(5\).
- **Exponent**: The exponent tells us how many times the base is used as a factor. It may also be referred to as the 'power'. In this case, \(-3\) is the exponent. The negative sign here indicates that we're dealing with a reciprocal.

Together, base and exponent define a power. The given expression \(5^{-3}\) states that the base \(5\) needs to be used in multiplication three times, but due to the negative exponent, it results in the reciprocal of its positive counterpart. This leads us to the result of \(\frac{1}{125}\). Simply put, \(5^{-3}\) equals \(1\) divided by \(5^3\).

Understanding this relationship is crucial when working with equations, as it helps us to comprehend the logic behind the numbers and their arrangement.

Logarithmic Form

Transforming a mathematical expression from an exponential form to a logarithmic form can initially seem complex. But with a clear approach, it's quite straightforward.

The exponential expression \(5^{-3} = \frac{1}{125}\) can be converted into a logarithmic form. The logarithmic form is a way to answer the question: 'To what power must we raise the base to get a certain number?' In essence, it's about switching focus from the result to the exponent.

- The base of the logarithm is the same as the base in the exponent form: \(5\).
- The expression \(\log_b{x} = y\) becomes \(\log_{5}{(\frac{1}{125})} = -3\).

This reads as 'logarithm base \(5\) of \(\frac{1}{125}\) equals \(-3\)'. Essentially, it's reversing the "base and exponent" logic by asking the question: "What power of \(5\) is \(\frac{1}{125}\)?" The answer, as calculated, is \(-3\).

Understanding how to convert between exponential and logarithmic forms is crucial for solving complex mathematical equations, and it's the foundation for more advanced topics in mathematics.

Equivalent Expressions

Recognizing the equivalence between different mathematical expressions is a powerful skill. It enables flexibility in problem-solving by letting us choose the most convenient form for our calculations.

In the exercise, we have two expressions:

• The exponential form: \(5^{-3} = \frac{1}{125}\).
• The logarithmic form: \(\log_{5}{(\frac{1}{125})} = -3\).

These two expressions, though appearing different at first glance, are indeed equivalent. They represent the same mathematical relationship from different perspectives. The exponential form shows how the base and exponent are used to achieve a specific result. Meanwhile, the logarithmic form focuses on determining the exponent when you have the base and the result.

This type of equivalency allows mathematicians and students alike to maneuver between different modes of expression to simplify calculations or to solve problems more effectively. Understanding both forms and how they relate can make navigating through complex mathematical landscapes much smoother.