Problem 5 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Express each equation in logarithmic form. $$\left(\frac{1}{3}\right)^{1}=\frac{1}{3}$$

Step-by-Step Solution

Answer

The logarithmic form of the given exponential equation \(\left(\frac{1}{3}\right)^{1}=\frac{1}{3}\) is: \[\log_{\frac{1}{3}}\left(\frac{1}{3}\right)=1\]

Step 1: Express the equation as a logarithm

To convert an exponential equation to logarithmic form, we proceed by reading the equation and pay attention to its base, exponent, and result. In our case, the base is \(\frac{1}{3}\), the exponent is \(1\), and the result is \(\frac{1}{3}\). The logarithmic form can be written as: \[ \log_{base}(result)=exponent \] Now let's substitute our values.

Step 2: Substitute the known values

Based on the given information, we have: - Base: \(\frac{1}{3}\) - Exponent: \(1\) - Result: \(\frac{1}{3}\) Plugging these values into the logarithmic form, we get: \[ \log_{\frac{1}{3}}\left(\frac{1}{3}\right)=1 \] So, the logarithmic form of the given exponential equation is: \[ \log_{\frac{1}{3}}\left(\frac{1}{3}\right)=1 \]

Key Concepts

Exponential Equations

When we deal with exponential equations, we are working with expressions where variables appear as exponents. A fundamental form of an exponential equation is where a number, known as the 'base', is raised to a power, called the 'exponent', and equals a certain 'result'.
Understanding how to solve exponential equations is crucial in mathematics, as they model scenarios of rapid growth or decay, such as population growth, radioactive decay, and interest calculations in finance.
An exemplar equation \( b^x = y \) can be translated into a logarithmic form to harness the power of logarithms for solving it. Here, \(b\) is the base, \(x\) the exponent, and \(y\) the result. The exercise given neatly illustrates a simple case where the base \( \frac{1}{3} \) is raised to the exponent of \(1\), yielding a result of \( \frac{1}{3} \) as well. These types of equations are crucial in advanced mathematics and their applications span across various scientific disciplines.

Logarithms

Switching Between Exponential and Logarithmic Form

Logarithms are incredibly useful in mathematics for simplifying the process of solving exponential equations. A logarithm answers the question, 'To what exponent must we raise a base to get a certain result?' Therefore, the logarithmic form is integral for solving exponential equations where the exponent is unknown.
In the example we've been given, translating \( \left(\frac{1}{3}\right)^1 = \frac{1}{3} \) to logarithmic form involves expressing the equation in the form \( \log_{base}(result) = exponent \). This transformation lays down a foundation for various mathematical methods that can solve for unknowns in more complex equations.

Understanding Logarithm Properties

Logarithms come with a set of properties that significantly ease solving equations: the product, quotient, and power rules are among these properties. They transform multiplication into addition, division into subtraction, and exponentiation into multiplication within a logarithmic context, which can simplify otherwise complex calculations.

Base and Exponent

The 'base' and 'exponent' are the fundamental components of any exponential equation. Our exercise \(\left(\frac{1}{3}\right)^1=\frac{1}{3}\) features a base of \(\frac{1}{3}\), which represents the number that is being multiplied by itself, and an exponent of \(1\), telling us how many times to multiply the base by itself.
In general terms, if we have \( b^n = y \), 'b' would be our base, 'n' the exponent, and 'y' the result.
This relationship is paramount when expressing equations in logarithmic form. In logarithms, the base of the logarithm corresponds to the base of the exponent in the original expression. It is a fundamental constant that ties exponential and logarithmic forms together, and mastering the concept of base and exponent is essential for understanding how to move between these two forms. This kind of fluency is important not only for academic purposes but also for practical problem-solving in sciences and engineering.