Problem 6 - A Graphical Approach to Precalculus with Limits

Question

For each statement, write an equivalent statement in logarithmic form. Do not use a calculator. $$\left(\frac{1}{100}\right)^{-2}=10,000$$

Step-by-Step Solution

Answer

The logarithmic form is \( \log_{\frac{1}{100}} 10,000 = -2 \).

Step 1: Understand the Exponential Form

The given equation is \( \left(\frac{1}{100}\right)^{-2} = 10,000 \). This is written in exponential form where the base is \( \frac{1}{100} \), the exponent or power is \( -2 \), and the result is \( 10,000 \).

Step 2: Apply the Definition of Logarithms

The definition of logarithms states that if \( a^b = c \), then the logarithmic form is \( \log_a c = b \). In this case, \( a = \frac{1}{100} \), \( b = -2 \), and \( c = 10,000 \).

Step 3: Convert to Logarithmic Form

Using the definition from Step 2, convert the given exponential statement to logarithmic form. It becomes \( \log_{\frac{1}{100}} 10,000 = -2 \).

Key Concepts

Exponential Form

The exponential form is a way of representing numbers using powers. Let's break it down:

• The **base** is the number being multiplied by itself, and the **exponent** tells us how many times the base is multiplied.
• In mathematical terms, an exponential expression is written as: \( a^b = c \). Here, \( a \) is the base, \( b \) is the exponent, and \( c \) is the result.

To understand a statement like \( \left(\frac{1}{100}\right)^{-2} = 10,000 \), we analyze each part:- **Base**: \( \frac{1}{100} \)- **Exponent**: \( -2 \)- **Result**: \( 10,000 \)Exponential form helps us express and understand large numbers and complex calculations more easily. By using the base and exponent together, it simplifies the representation of repeated multiplication. This is particularly useful in fields like science and engineering, where very large or small values are common.

Base and Exponent

The concepts of base and exponent are fundamental to understanding exponential expressions. Here's what each term means:

• The **base** is the 'big' number, or the foundational value, that is repeatedly multiplied.
• The **exponent** (also known as the power) tells us how many times the base is used as a factor in the multiplication.

For the equation \( \left(\frac{1}{100}\right)^{-2} = 10,000 \):- The **base** is \( \frac{1}{100} \), a fractional representation indicating how something very small can grow.- The **exponent** is \(-2\), which indicates that the base is to be divided instead of multiplied, due to the negative sign. In this case, it means the reciprocal (or inverse) of the base squared.The negative exponent \(-2\) means rewriting the base as its reciprocal, and raising it to a positive power, specifically \((\frac{100}{1})^2 = 10,000\). Understanding the interaction between base and exponent is key in fields like algebra and calculus, allowing for manipulation and transformation of equations to solve for unknowns.

Convert to Logarithmic

Converting an exponential equation to logarithmic form is a fundamental skill in algebra. It involves switching the base, exponent, and the result in a new equation form. Here's how it's done:- The general function of exponentials \( a^b = c \) converts to logarithmic form as \( \log_a c = b \).For the equation \( \left(\frac{1}{100}\right)^{-2} = 10,000 \), we apply this conversion:

• **Base**: \( \frac{1}{100} \)
• **Exponent**: \( -2 \)
• **Result**: \( 10,000 \)

By plugging these into the logarithmic form, we get \( \log_{\frac{1}{100}} 10,000 = -2 \).This transformation makes it possible to solve for the exponent when both the base and the result are known. This is an essential part of calculus and analysis, where understanding the growth rates and scaling of functions is crucial. Mastering this conversion helps unravel complex problems and provides insights into the world of exponential growth, decay, and scalability.