Problem 8 - College Algebra and Calculus: An Applied Approach

Question

Use the definition of a logarithm to write the equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3$$7^{3}=343\)

Step-by-Step Solution

Answer

The logarithmic form of the equation \(7^{3}=343\) is \(\log _{7} 343=3\).

Step 1: Identify the Base, Exponent, and Result

In the given equation \(7^{3}=343\), the base is 7, the exponent is 3, and the result or value is 343.

Step 2: Understand the Logarithmic Form

A logarithmic form of an equation is written as \(\log _{base} \: result = exponent\). This means the logarithm of the 'result' to the 'base' is equal to the 'exponent'.

Step 3: Write the Equation in Logarithmic Form

Replace the 'base', 'result', and 'exponent' in logarithmic form with 7, 343, and 3, respectively. So the logarithmic form of the equation \(7^{3}=343\) is \(\log _{7} 343=3\).

Key Concepts

Base and Exponent

When diving into the world of exponents, two key players are involved: the base and the exponent. Let's break these down with the equation provided in the exercise: \(7^3 = 343\). Here:

• The base is the number 7. Think of the base as the foundation or the number that is being multiplied.
• The exponent is 3. This tells us how many times the base is used as a factor in the multiplication.
• Together, the equation implies that 7 is multiplied by itself 3 times: \(7 \times 7 \times 7 = 343\).

Learning this concept helps us understand the exponential "power" we are dealing with. Moreover, grasping the basics of a base and exponent is crucial for understanding how we transition to logarithms.

Logarithmic Form

Logarithms can seem tricky at first, but they are simply another way to express a relationship between numbers, much like exponents. A logarithm answers the question: "To what exponent must the base be raised, to equal a given number?"

In the example \(7^3 = 343\), the logarithmic form expresses this same equation differently. It is written as \(\log_{7} 343 = 3\).

• The base remains 7, indicating the number we are repeatedly multiplying.
• The result, formerly known as the value of the exponential expression, is 343.
• And the exponent, the power indicating the number of repetitions of multiplication, is still 3.

The essential idea here is that a logarithm shows us how many times the base must be multiplied to reach a particular number. This is fundamental when unraveling exponential equations.

Mathematical Equations

Equations are the backbone of mathematics, acting as statements that assert the equality of two expressions. Let's understand how this is applied to both exponential and logarithmic equations.

In exponential form, our statement \(7^3 = 343\) is straightforward. It tells us the product of multiplying 7 by itself twice more to yield 343. But in logarithmic form, \(\log_{7} 343 = 3\), the equation subtly shifts in meaning to express the same relationship: it states that 3 is the power to which the base 7 must be raised to result in 343.

• There are clear rules and relationships within equations that allow us to transform one form into another, via fundamental operations and understanding.
• Grasping this fluidity between forms—exponential and logarithmic—empowers us in solving various mathematical problems.
• Both forms serve specific purposes, but one’s comprehension of equations and their transformations is integral to mastering math.

By appreciating both methodologies, students can navigate through problems with ease and confidence.