Problem 126 - Elementary and Intermediate Algebra

Question

Show that \(e^{x \ln a}=a^{x}\).

Step-by-Step Solution

Answer

Using the power rule and the identity \( e^{\ln a} = a \), we show \( e^{x \ln a} = a^x \).

Step 1: Express the Exponential

Start by expressing the left-hand side of the given equation: \( e^{x \ln a} \). This is an exponential function with base \( e \) and exponent \( x \ln a \).

Step 2: Use the Power Rule

According to the power rule of exponents, \( e^{x \ln a} = (e^{\ln a})^x \). This rule states that raising an exponential with a logarithm in the exponent can be rewritten by isolating the base of the logarithm.

Step 3: Simplify Using Logarithmic Identity

Recall the identity \( e^{\ln a} = a \). Substituting back, we have \((e^{\ln a})^x = a^x\).

Step 4: Left and Right Side Comparison

We have reduced the expression on the left-hand side to \( a^x \), which matches the right-hand side of the original equation. Thus, \( e^{x \ln a} = a^x \) is shown to be true.

Key Concepts

Power Rule of Exponents

The power rule of exponents is a fundamental concept in mathematics that simplifies how we deal with exponents, or powers. When you have an expression of the form \( (a^m)^n \), the power rule allows you to combine the exponents. The result of applying the power rule is given by:

• \( (a^m)^n = a^{m imes n} \)

Let's break it down a little more. Here, \(a\) is the base, and \(m\) and \(n\) are the exponents. The rule states you should multiply \(m\) and \(n\) together to get a simplified result.
In the context of our original problem \(e^{x \ln a} = (e^{\ln a})^x\), using the power rule helps us rearrange the exponent by isolating the base of the logarithm, making it much easier to simplify.
For instance, recognizing that \(e^{x \ln a}\) can be rewritten as \((e^{\ln a})^x\) illustrates the power rule. This steps sets the stage for further simplification by using another math concept called a logarithmic identity.

Logarithmic Identities

Logarithmic identities are powerful tools for simplifying expressions that involve logarithms. One of the most essential identities you'll encounter is \( e^{\ln a} = a \).
This identity allows us to "cancel" the exponential and the logarithmic function when they are inverse operations of each other. Here’s how it works more closely:

• The natural logarithm \( \ln a \) is the inverse of the exponential function \( e^x \).
• When \( e \), the base of natural logarithms, is raised to \( \ln a \), it simplifies directly to \(a\).

In the context of our exercise, we use this identity to simplify the expression \((e^{\ln a})^x\) to \(a^x\). This step is crucial because it shows that the exponential expression on one side of our original equation can be made to exactly match the power on the other side. By knowing how these identities work, you can transform and simplify complex logarithmic expressions.

Mathematical Proofs

A mathematical proof is a logical argument that demonstrates why a particular statement is true. It uses a sequence of logical deductions based on axioms, definitions, and previously proven statements. The goal is to confirm beyond any doubt that the proposition holds true in all possible cases.
In our exercise, the proof involves showing the equivalence of \( e^{x \ln a} \) and \( a^x \). For this:

• We start with the original expression \( e^{x \ln a} \).
• Use the power rule of exponents to rearrange the powers in a recognizable form.
• Apply logarithmic identities to simplify the expression further, achieving \( a^x \).

By comparing both sides and seeing they are indeed equal, we have provided a proof. Proofs like this one strengthen our understanding of mathematical principles, showing how different rules interact. So each time you engage with a proof, you're effectively discovering why math works the way it does!