Problem 18 - Algebra for College Students

Question

Write each logarithmic statement in exponential form. For example, \(\log _{2} 8=3\) becomes \(2^{3}=8\) in exponential form. $$ \log _{5}\left(\frac{1}{125}\right)=-3 $$

Step-by-Step Solution

Answer

\( 5^{-3} = \frac{1}{125} \)

Step 1: Understanding Logarithmic Format

The given statement is \( \log _{5}\left(\frac{1}{125}\right)=-3 \). This is in logarithmic form where 5 is the base, \( \frac{1}{125} \) is the result, and -3 is the exponent.

Step 2: Rewriting in Exponential Form

In exponential form, the base raised to the power of the exponent equals the result. Therefore, \( \log _{5}\left(\frac{1}{125}\right)=-3 \) becomes \( 5^{-3} = \frac{1}{125} \).

Key Concepts

Logarithmic Statements

Logarithms provide a powerful way to express exponential relationships. In a logarithmic statement, you have three main components: a base, an exponent (also known as the logarithm), and the result of the power operation.
For instance, in the statement \( \log_{5}\left(\frac{1}{125}\right)=-3 \), the base is 5, the exponent is -3, and the result is \( \frac{1}{125} \).
This notation tells us that when the base 5 is raised to the exponent -3, it equals \( \frac{1}{125} \). By understanding logarithmic statements, you are essentially unlocking the relationship between these three components. Each part plays a vital role in the equation's balance and offers insights into the power and scale of numbers.

Base and Exponent

The base and exponent are central to both logarithmic statements and their exponential form. In mathematical terms, the base refers to the number repeatedly multiplied by itself - the building block of the power operation.
For instance, in \( \log_{5}\left(\frac{1}{125}\right)=-3 \), the base is 5. The exponent, represented by -3, indicates how many times the base is multiplied, factoring in whether it's a positive or negative power.

• If the exponent is positive, you multiply the base by itself that many times.
• If the exponent is negative, you take the reciprocal of the base raised to the absolute value of the exponent.

Thus, an understanding of bases and exponents allows you to seamlessly navigate between logarithmic and exponential forms, working fluidly between these two frameworks to solve mathematical problems.

Converting Logarithms to Exponents

Converting a logarithmic statement to its exponential form is a straightforward process once you grasp the basic concepts. The key is recognizing that a logarithm represents an exponent in disguise.
To convert, you start with the original logarithmic statement: \( \log_{5}\left(\frac{1}{125}\right)=-3 \). The goal is to express this in exponential terms.

• Identify the base (5), the exponent (-3), and the result \( \frac{1}{125} \).
• Using the format \( b^e = r \), write the exponential form as \( 5^{-3} = \frac{1}{125} \).

This step ensures that each part of the logarithmic statement is properly converted into its exponential counterpart.

Mathematical Notation

Mathematical notation is the language in which mathematics is communicated. It's a universal system of symbols and principles designed to convey mathematical concepts succinctly and precisely.
In logarithms and exponents, clear and correct notation is essential. For instance, the expression \( \log_{5} \) clearly indicates we are dealing with a logarithm of base 5. Similarly, \( 5^{-3} \) shows an exponentiation operation with base 5 raised to a power of -3.

• Symbols like \( \log \), \( ^ \), and numeric indices are utilized to communicate complex ideas with brevity.
• Accurate notation ensures that the meaning is universally understood and maintains consistency across mathematical discourse.

Learning to interpret and employ mathematical notation fluently is crucial, as it not only aids in problem-solving but also in effectively communicating mathematical ideas.