Problem 7 - Contemporary Precalculus

Question

Translate the given logarithmic statement into an equivalent exponential statement. $$\log 750=2.88$$

Step-by-Step Solution

Answer

Question: Convert the logarithmic statement \(\log 750=2.88\) into an equivalent exponential statement. Answer: The equivalent exponential statement is \(10^{2.88}=750\).

Step 1: Identify the base

In the given logarithmic statement, the base is not explicitly shown. However, we can understand that by default, logarithms have a base of 10 when it is not specified. Therefore, the base, \(b\), is 10.

Step 2: Identify the exponent and the result

Using the given logarithmic statement \(\log 750=2.88\), we can identify the exponent "\(c\)" to be \(2.88\) and the result "\(a\)" of the exponential statement to be \(750\).

Step 3: Write the equivalent exponential statement

Using the relationship \(\log_b a = c\) is equivalent to \(b^c = a\), and the base of 10, we can write the equivalent exponential statement: $$10^{2.88}=750$$

Key Concepts

Logarithms

Logarithms are an integral part of mathematics, particularly when dealing with multiplicative processes and growth patterns. They are the inverse operations to exponentiation, which means they undo what exponentiation does. A logarithm tells us the exponent required for a number, known as the base, to reach another number.

A standard logarithmic statement is formatted as \(\log_b a = c\), where \(\b\) is the base, \(\b\) is the result, and \(c\) is the exponent. If the base is not specified, as in the expression \(\b\), it is understood to be 10, which is known as the 'common logarithm'.

To grasp the concept of logarithms, it is essential to recognize that \(\log_b a = c\) is asking the question: 'To what exponent must we raise the base \(\b\) to obtain the number \(\b\)?' This interrelationship between logarithms and exponents lays the foundation for converting between logarithmic and exponential forms.

Exponential Statements

Exponential statements express the relationship between a base raised to an exponent resulting in a certain number. Represented as \(\b^c = a\), where \(\b\) is the base, \(\b\) is the exponent, and \(\b\) is the result or output number. This form is extremely useful in a multitude of mathematical scenarios, such as compound interest calculations, population growth models, and radioactive decay, among others.

In our exercise, the move from a logarithmic statement to an exponential statement involved identifying the base (10, by default for a common logarithm), the exponent (2.88), and the result (750). This conversion is critical not just as an academic exercise but also for practical applications, like solving equations in calculus or understanding the behavior of log-based functions in computer science.

Exponential Functions

Exponential functions are mathematical functions of the form \(\b(x) = a \b^{x}\), where the base \(\b\) is a positive real number. Unlike linear functions with a constant rate of change, exponential functions grow by common factors over equal intervals. This means that as \(\b\) increases, the function's rate of growth increases or decreases at an exponential rate, depending on whether the base is greater than or less than 1.

In our textbook exercise, converting \(\b 750=2.88\) to its equivalent exponential form \(\b^{2.88} = 750\) underscores the concept of an exponential function. This is because it demonstrates how a given input, like 2.88, affects the output, 750, in an exponential relationship. Understanding exponential functions is quintessential in fields such as economics, biology, and physics where exponential growth or decay phenomena are commonplace.