Problem 14 - Intermediate Algebra

Question

For Problems \(11-20\), write each of the following in exponential form. For example, \(\log _{2} 8=3\) becomes \(2^{3}=8\) in exponential form. $$ \log _{5} 25=2 $$

Step-by-Step Solution

Answer

The exponential form is \(5^{2} = 25\).

Step 1: Understand the Problem Statement

We are given a logarithmic equation \(\log_{5} 25 = 2\) and need to rewrite it in exponential form.

Step 2: Apply the Definition of Logarithms

Recall that the logarithmic form \(\log_{b}(a)=c\) is equivalent to the exponential form \(b^{c}=a\). In this problem: \(b=5\), \(a=25\), and \(c=2\).

Step 3: Substitute Values into the Exponential Form

Using the values identified in Step 2, substitute into the exponential form: \(5^{2} = 25\).

Step 4: Verify the Exponential Expression

Calculate \(5^{2}\) to confirm it equals 25, which confirms that the exponential expression \(5^{2}=25\) is correct.

Key Concepts

Logarithms

Logarithms are mathematical operations that help us discern relationships between numbers through exponents. In simple terms, a logarithm answers the question: "To what exponent must a given base be raised, to yield a particular number?" For example, in the expression \(\log_{5} 25 = 2\), the logarithm tells us that the base 5 must be raised to the power of 2 to get 25. Logarithms are inverse operations of exponentials, much like how subtraction is the inverse of addition. This property allows us to solve for unknown exponents when a number and its base are known.
Understanding logarithms requires familiarity with their basic form: \(\log_{b} a = c\). This notation reads as "the logarithm of \(a\) with base \(b\) equals \(c\)." Here, \(b\) is the base, \(a\) is the result of raising \(b\) to some power, and \(c\) is the exponent to which \(b\) is raised.

Base of Logarithm

The base of a logarithm is a fundamental part of its structure and essential to understanding exponential relationships. In a logarithmic expression, such as \(\log_{5} 25 = 2\), the number 5 is the base. The base can be any positive number except 1, because raising 1 to any power will always result in 1. In general, a logarithm \(\log_{b} a = c\) can be interpreted as: what power must the base \(b\) be raised to, in order to produce \(a\)?

• The identity \(\log_{b} 1 = 0\) holds because any non-zero base raised to the power 0 is 1.
• Common bases include 10 (common logarithm) and \(e\) (natural logarithm), but in logarithmic forms like in our exercise, any number, such as 5 here, can be a base.
• Understanding the base is crucial as it determines the framework within which exponential expressions are constructed or deconstructed using logarithms.

Recognition of the base helps in translating a logarithmic expression back into its exponential form as we did in the solution: \(5^{2} = 25\).

Exponential Expressions

Exponential expressions show the process of raising numbers to powers. In math, these expressions are concise ways to represent repeated multiplication. For instance, when you see \(5^{2}\), it means multiplying 5 by itself, resulting in 25. Converting a logarithmic expression, like \(\log_{5} 25 = 2\), into its exponential form, involves rearranging the expression to exhibit this multiplication: \(5^{2} = 25\).
When dealing with exponential expressions:

• The base appears as the factor being multiplied.
• The exponent shows how many times the base is used in the multiplication.
• Such expressions are crucial for representing large computations succinctly.
• In science, exponential expressions are often used to represent growth, decay, scales, or units like in scientific notation.

Mastering exponential expressions can make complex mathematical concepts easier to understand and solve, as they transform multiplication into an approachable format through simple notation.