Problem 23 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ 7^{\log _{7} 2 x} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 2x \).

Step 1: Understanding the Problem

We need to simplify the expression \( 7^{\log_{7}(2x)} \). The goal is to reduce it to its simplest form.

Step 2: Applying the Power Rule of Logarithms

Notice that \( 7^{\log_{7}(2x)} \) is of the form \( b^{\log_{b}(a)} \). By the power rule of logarithms, \( b^{\log_{b}(a)} = a \). Here, \( b = 7 \) and \( a = 2x \).

Step 3: Simplifying Using the Power Rule

Applying the power rule, the expression \( 7^{\log_{7}(2x)} \) simplifies directly to \( 2x \). This is because the base of the power and the base of the logarithm are the same.

Key Concepts

Simplification

The concept of simplification is all about making an expression easier to work with. Sometimes, mathematical expressions might look complex at first, but there are rules and techniques to transform them into a much simpler form. The goal is to make calculations or further manipulations of the expression straightforward and clear.

In the context of logarithmic functions, simplification can involve using properties of logarithms to reduce expressions. For example, when you see an expression like \( 7^{\log_{7}(2x)} \), initially, it might seem a bit complicated. But with the right simplification techniques, such as recognizing patterns or applying mathematical rules, you can break it down to something much easier to manage.

Keep in mind that simplification not only helps in finding solutions efficiently but also aids in enhancing your understanding of the relationships within the problem.

In the given exercise, recognizing that \( 7^{\log_{7}(2x)} \) follows a recognizable pattern allows us to simplify it quickly and effectively.

Power Rule of Logarithms

The Power Rule of Logarithms is a crucial concept when dealing with expressions involving logarithms and exponents. This rule states that for a given base \( b \) and a positive number \( a \), the expression \( b^{\log_{b}(a)} \) simplifies to \( a \). This is because of the inherent property of logarithms that tell us a logarithm can undo exponentiation when the base is the same.

• This means if you have an expression where the base of an exponent is the same as the base of its logarithm, you can directly switch to the argument of the logarithm.
• For example, in our expression \( 7^{\log_{7}(2x)} \), the base of the exponent (7) matches the base of the logarithm (also 7).
• Therefore, according to the power rule, this simplifies to just \( 2x \).

The Power Rule is powerful because it allows you to eliminate seemingly complex operations into straightforward expressions simply by recognizing matching bases.

Understanding this rule helps you quickly resolve problems without excessive computations, saving both time and effort.

Exponential Functions

Exponential functions are a fundamental concept in mathematics, notable for their constant proportional growth rate. These functions are expressed generally in the form \( y = b^x \), where \( b \) is the base of the exponential function, and \( x \) is the exponent.

• A key property of exponential functions is their rapid growth or decay. As \( x \) increases, the value of \( y \) changes exponentially, rather than linearly.
• In the context of our exercise, the expression \( 7^{\log_{7}(2x)} \) involves an exponential function where the base of the exponent is the same as the base of the logarithm.

Combining exponential functions and logarithms often reveals relationships between growth processes and their inverses. In our exercise, once the base of the exponential matches the base of the logarithm, they effectively "cancel out," highlighting the defining feature of exponentials as the inverse function of logarithms.

This elegant property makes exponential and logarithmic functions incredibly useful in various fields, from compound interest calculations in finance to growth models in biology, showcasing their critical role in describing real-world phenomena.