Convert the following expressions to the indicated base.


$a^{\frac{1}{\ln a}}$ using basa e, for $a>0$ and $a\ne 1$

Verified step by step guidance

1. Start by recognizing that the expression \( a^{\frac{1}{\ln a}} \) can be rewritten using the property of exponents and logarithms. We know that \( a^x = e^{x \ln a} \).
2. Apply this property to the given expression: \( a^{\frac{1}{\ln a}} = e^{\frac{1}{\ln a} \cdot \ln a} \).
3. Simplify the exponent: \( \frac{1}{\ln a} \cdot \ln a = 1 \).
4. Substitute back into the expression: \( e^{1} \).
5. Conclude that the expression simplifies to \( e \).>

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form of f(x) = a^x, where 'a' is a positive constant. These functions are characterized by their rapid growth or decay, depending on the base. Understanding how to manipulate and convert between different bases is crucial for solving problems involving exponential expressions.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base 'e', where 'e' is approximately equal to 2.71828. It is the inverse function of the exponential function with base 'e'. The natural logarithm is essential for converting exponential expressions into a more manageable form, particularly when dealing with expressions like a^(1/ln(a)).

Change of Base Formula

The change of base formula allows for the conversion of logarithms from one base to another. It states that log_b(a) = log_k(a) / log_k(b) for any positive k. This concept is particularly useful when converting expressions to a specific base, such as converting a^x to base 'e' in the given problem.