Problem 2 - Algebra 2

Question

Write each equation in logarithmic form. \(7^{-2}=\frac{1}{49}\)

Step-by-Step Solution

Answer

\(\log_7 \frac{1}{49} = -2\)

Step 1: Identify the Exponential Equation

The given equation is an exponential equation: \(7^{-2} = \frac{1}{49}\). In this equation, 7 is the base, -2 is the exponent, and \(\frac{1}{49}\) is the result.

Step 2: Understand Logarithmic Form

The logarithmic form for an equation \(b^x = y\) is \(\log_b y = x\), where \(b\) is the base, \(y\) is the result, and \(x\) is the exponent in the original exponential form.

Step 3: Convert the Given Equation

Using the logarithmic form, \(7^{-2} = \frac{1}{49}\) can be rewritten by identifying the base, the result, and the exponent as \(\log_7 \frac{1}{49} = -2\).

Step 4: Write the Logarithmic Form

The given exponential equation \(7^{-2} = \frac{1}{49}\) is written in logarithmic form as \(\log_7 \frac{1}{49} = -2\).

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where a constant base is raised to a variable exponent. These equations appear in various scientific fields, including physics, computer science, and finance. The general form of an exponential equation is given by \(b^x = y\), where \(b\) is the base, \(x\) is the exponent, and \(y\) is the result.

Exponential equations are essential because they model growth and decay processes, such as population growth or radioactive decay. The base \(b\) determines the growth rate. If \(b > 1\), the function represents exponential growth; if \(0 < b < 1\), it represents exponential decay.

Understanding exponential equations helps in converting them into logarithmic form. This conversion is particularly useful for solving equations where the unknown is an exponent, as it simplifies the process by transforming multiplication into addition.

Logarithms

Logarithms are the inverse operations of exponentiations, acting much like how subtraction is the inverse of addition. They play a key role in simplifying complex exponential equations by converting them into a more manageable form.

A logarithm answers the question: "To what exponent must we raise the base \(b\) to obtain a certain number \(y\)?" Hence, the expression \(\log_b y = x\) directly corresponds to the exponential equation \(b^x = y\). Here:

• \(b\) is the base of the logarithm.
• \(y\) is the number you want to find the logarithm of.
• \(x\) is the exponent or the logarithm's value.

The logarithmic form makes calculations easier, as it transforms multiplication into addition and division into subtraction, which are often simpler to perform.

Logarithms have practical applications across various domains, such as in measuring the intensity of sound (decibels), defining the pH level in chemistry, and in calculating compound interest in finance.

Mathematical Notation

Mathematical notation serves as a universal language that enables precise and clear communication of concepts and methods within mathematics and related fields. It is a concise and systematic way to represent mathematical ideas, procedures, and problems.

For exponential equations and logarithms, specific symbols and notation provide clarity and prevent ambiguity:

• \(b\), \(x\), and \(y\) are typically used to signify base, exponent, and result, respectively, in exponential equations.
• In logarithms, \(\log_b y = x\) conveys the relationship between base, result, and the exponent.

Proper use of mathematical notation is crucial when converting equations from exponential to logarithmic form or vice versa, as it ensures all steps are logically followed and accurately recorded.

Mathematical notation not only aids in solving exercises but also in writing proofs, documenting steps in calculations, and sharing mathematical insights efficiently across educational resources and publications.