Problem 33 - Technical Mathematics with Calculus

Question

Simplify each expression. $$2^{\log _{2} 3 y}$$

Step-by-Step Solution

Answer

The simplified expression is 3y.

Step 1: Apply algebraic rules

Use properties of exponents, radicals, fractions, or algebraic identities to simplify.

Step 2: State the result

The simplified expression is The simplified expression is 3y..

Key Concepts

Logarithmic Relationships

Understanding logarithmic relationships is essential when dealing with exponential expressions. In essence, a logarithm answers the question: to what exponent must we raise a certain base to obtain a given number? This interplay between logarithms and exponents forms the foundation of many algebraic operations involving these two concepts.

For instance, the logarithmic function that is the inverse of the exponentiation process can be written as \(\log_b(x) = y\) implies that \(b^y = x\). This is particularly useful in simplifying expressions where a base is raised to a power that is a logarithm of a number using that same base, as seen in the original exercise \(2^{\log_{2}3y}\).To visualize this relationship, imagine any number \(x\) that we reach by taking a base \(b\) and raising it to the power of \(y\); conversely, if we know the base \(b\) and the number \(x\), we can find out what exponent \(y\) was used through the logarithm \(\log_b(x)\). This circular relationship is a key concept in simplifying expressions involving exponents and logarithms.

Exponential and Logarithmic Properties

Exponential and logarithmic properties provide a set of rules that guide the simplification of expressions involving exponents and logarithms. Understanding these properties makes it easier to solve complex algebraic equations and can lead to more intuitive insights into the behavior of logarithmic and exponential functions.

One of the most powerful properties when dealing with expressions that combine both concepts is the inverse nature of exponential and logarithmic functions: logarithms are the inverse operations of exponentiation. This means that for any base \(b\) and exponent \(y\), if you exponentiate \(b\) to \(y\) to get \(x\), then \(\log_b(x)\) will equal \(y\). This reciprocal relationship allows for simplifications such as converting an exponential expression into a logarithmic form and vice versa.Furthermore, the rules regarding the manipulation of exponents, such as the power of a product, power of a quotient, or power of a power, have their corresponding logarithmic properties. For example, the logarithm of a product can be written as the sum of the logarithms \(\log_b(mn) = \log_b(m) + \log_b(n)\). These properties are indispensable tools for manipulating and simplifying expressions containing exponents and logarithms.

Fundamental Property of Logarithms

The fundamental property of logarithms is a cornerstone of algebra that allows us to simplify expressions like \(2^{\log_{2}3y}\) in the original exercise. It states that for any positive number \(x\), a base \(b\), where \(b\) is positive and \(beq1\), the expression \(b^{\log_b(x)}\) simplifies to \(x\). This happens because, by definition, \(\log_b(x)\) is the exponent to which \(b\) must be raised to yield \(x\).In practical terms, when we encounter an expression where a base is raised to the logarithm of a number with the same base, we can know right away that the base and the logarithm 'cancel each other out'. The rationale behind this is that since the logarithm is giving us the exponent needed to reach a certain number using our base, raising the base to this logarithm is an unnecessary step—we can go directly to the number. In the original problem, this means that the expression \(2^{\log_{2} 3y}\) simplifies directly to \(3y\), because \(2\) raised to the power that results in \(3y\) is, unsurprisingly, \(3y\) itself.