Question

Convert the following expressions to the indicated base. \(a^{1 / \ln a}\) using base \(e,\) for \(a>0\) and \(a \neq 1\)

Short answer

Question: Convert the expression \(a^{1/\ln(a)}\) to base \(e\). Answer: The expression \(a^{1/\ln(a)}\) converted to base \(e\) is \(e\).

Step-by-step solution

Write the given expression using the base \(e\) logarithm (ln)

Write the given expression \(a^{1/\ln(a)}\) as the exponential in base \(e\): \(\exp(\ln(a^{1/\ln(a)}))\).

Apply log properties on the expression

Using the property of logarithms, \(\log(a^b)=b\log(a),\) we express the exponent inside the logarithm: \(\exp\left(\frac{1}{\ln(a)} \cdot \ln(a)\right)\).

Simplify the expression

As we can see, the exponent inside the exponential function is now a fraction with the same numerator and denominator, so they cancel each other out: \(\exp\left(\frac{1}{\ln(a)} \cdot \ln(a) \right)=\exp(1)\). Hence, the expression \(a^{1/\ln(a)}\) converted to base \(e\) is \(\exp(1) = e.\)

Key concepts

Exponential Functions

Exponential functions are fundamental mathematical expressions where the variable is in the exponent. In these functions, a constant base is raised to the power of a variable exponent. They're often written as \(b^x\), where \(b\) is the base and \(x\) is the exponent. Exponential functions are prevalent in various fields from science to finance because they model growth and decay processes, such as population growth or radioactive decay. They are known for their unique characteristic where their rate of change is proportional to their value, leading to rapid increases or decreases.

Key properties include:

• They are continuous and increase or decrease monotonically depending on whether the base is greater than or less than one.
• The derivative of \(e^x\) is itself, making \(e\) a natural base for exponential functions due to its unique properties in calculus.

Understanding how to manipulate and interpret exponential functions is crucial for solving many mathematical problems and understanding real-world phenomena.

Change of Base Formula

The change of base formula is a handy mathematical tool used for converting logarithms from one base to another. It’s particularly useful since most calculators and computational tools primarily support base 10 (common logarithms) and base \(e\) (natural logarithms).

The formula is given by:

• \(\log_b a = \frac{\log_c a}{\log_c b}\)

This equation means that the logarithm of \(a\) with base \(b\) can be expressed as a fraction of two logarithms in a new base \(c\). The top of the fraction is the logarithm of \(a\) in the new base, and the bottom is the logarithm of \(b\) in the new base.

To use the change of base formula effectively:

• Select a convenient base, often base 10 or \(e\), as these are commonly available in calculators.
• Calculate the numerator, which is the logarithm of \(a\) in the new base.
• Calculate the denominator, which is the logarithm of \(b\) in the new base.
• Divide the results to find the equivalent logarithm in the desired base.

Mathematical Proof

Mathematical proof is a logical argument that verifies the truth of a mathematical statement, based on previously established theorems and axioms. It's a fundamental concept in mathematics that ensures concepts are thoroughly vetted with absolute certainty. The process typically involves deriving a statement directly from known facts, using a series of logical deductions.

A proof consists of:

• Hypothesis: Starting assumptions or given conditions of the problem.
• Logical Progression: Methodical steps which connect the hypothesis to the conclusion.
• Conclusion: The final statement or theorem you've proven to be true.

In the exercise above, proving \(a^{1/\ln(a)} = e\) utilized properties of logarithms and exponentials through:

• Recognizing the use of the base \(e\) logarithm to convert expressions.
• Applying logarithmic properties to simplify complex expressions.
• Careful calculation where inverse operations cancel each other out.

Understanding proofs not only deepens comprehension of specific topics but also enhances critical thinking and problem-solving skills.