Question

Express \(3^{x}, x^{\pi},\) and \(x^{\sin x}\) using the base \(e\).

Short answer

Question: Rewrite the expressions \(3^{x}, x^{\pi},\) and \(x^{\sin x}\) using base \(e\). Answer: The expressions rewritten using base \(e\) are \(3^{x} = e^{x \ln 3}, x^{\pi} = e^{\pi \ln x},\) and \(x^{\sin x} = e^{\sin x \ln x}\).

Step-by-step solution

Rewrite first expression using base \(e\)

We have the expression \(3^{x}\). Using the property \(a^{b} = e^{b \ln a}\), we can rewrite it as follows: \(3^{x} = e^{x \ln 3}\)

Rewrite second expression using base \(e\)

Now, for the expression \(x^{\pi}\). Using the property \(a^{b} = e^{b \ln a}\), we can rewrite it as follows: \(x^{\pi} = e^{\pi \ln x}\)

Rewrite third expression using base \(e\)

Finally, for the expression \(x^{\sin x}\). Using the property \(a^{b} = e^{b \ln a}\), we can rewrite it as follows: \(x^{\sin x} = e^{\sin x \ln x}\)

Final answer using base \(e\)

Therefore, the given expressions expressed using the base \(e\) are: \(3^{x} = e^{x \ln 3}\\ x^{\pi} = e^{\pi \ln x}\\ x^{\sin x} = e^{\sin x \ln x}\)

Key concepts

Base e Conversion

Converting expressions into base \( e \) is a useful technique in mathematics, particularly in calculus and analysis. Base \( e \), also known as Euler's number, is approximately 2.71828. It is the base of the natural logarithm and is a fundamental constant in mathematics.
When converting an expression like \( a^b \) into base \( e \), we use the formula:

• \( a^b = e^{b \ln a} \)

This is possible because any number raised to a power can be expressed as a power of \( e \). The conversion happens by realizing that raising a number to a power means scaling the exponent of \( e \) using the natural logarithm.
For instance, in the exercise, \( 3^x \) can be rewritten as \( e^{x \ln 3} \). This conversion helps in simplifying differentiation and integration tasks, making it easier to deal with complex exponents.

Natural Logarithm

The natural logarithm, denoted \( \ln \), is the logarithm to the base \( e \). It is the inverse operation of exponentiation with base \( e \). This means that if \( e^y = x \), then \( \ln x = y \). Understanding the natural logarithm is crucial for converting powers into expressions that involve base \( e \).
In our conversion formula \( a^b = e^{b \ln a} \), the \( \ln a \) component computes the power that \( e \) must be raised to obtain \( a \). Hence, \( \ln \) translates multiplication in the power into addition in the exponent, matching the properties of exponential functions.
Some key properties of natural logarithms include:

• \( \ln(ab) = \ln a + \ln b \)
• \( \ln(a^b) = b \ln a \)
• \( \ln(1) = 0 \)

These properties are often used to simplify complex exponential expressions.

Exponentiation

Exponentiation is the mathematical operation involving two numbers, the base and the exponent. It is written as \( a^b \) where \( a \) is the base and \( b \) is the exponent. This operation signifies the repeated multiplication of the base \( a \).
To convert expressions such as \( x^\pi \) and \( x^{\sin x} \) to base \( e \), we use the property \( a^b = e^{b \ln a} \). This transformation is essential when dealing with calculus problems as it can simplify the differentiation and integration of functions with variable bases and exponents.
Here are some basic rules of exponentiation:

• \( a^0 = 1 \) for any \( a eq 0 \)
• \( a^1 = a \)
• \((a^m)^n = a^{m \cdot n}\)

Expressing expressions in terms of \( e \) is often necessary for solving complex mathematical problems efficiently.