Problem 95 - College Algebra

Question

Evaluate or simplify each expression without using a calculator. $$ \ln e^{9 x} $$

Step-by-Step Solution

Answer

The simplified expression is \(9x\).

Step 1: Apply logarithmic identities

Use the property of logarithms that states \( \ln e^{x} = x \). Therefore, based on this rule, \( \ln e^{9x} = 9x \).

Step 2: Simplify

Now the expression is simplified to \(9x\). There are no further simplification steps needed, so this is the final result.

Key Concepts

Natural Logarithm

Understanding the natural logarithm is essential for students tackling a range of mathematical problems. The natural logarithm, denoted as ln, is the logarithm to the base e, where e is an irrational and transcendental number approximately equal to 2.71828. It often appears in various fields of science and engineering, reflecting continuous growth or decay rates.

In the exercise given, ln is applied to an exponential function, specifically e raised to the power of 9x. The fundamental property that \( \text{ln} e^x = x \) allows us to simplify this expression directly to 9x. This step is crucial because it demonstrates how logarithms permit the 'unpacking' of the exponent, revealing the power to which e is raised - a technique frequently used to solve equations involving exponential growth or decay.

Exponential Functions

Exponential functions are a class of mathematical expressions where a constant base is raised to a variable exponent. The general form is f(x) = b^x, where b is the base, and x is the exponent. In the context of our problem, the base is the constant e, which is a special case known as the natural exponential function, represented by e^x.

Exponential functions are known for their distinct property of representing processes that change at rates proportional to their current value, which makes them incredibly useful for modeling population dynamics, compound interest, and radioactive decay, among other phenomena. In algebra, these functions are often manipulated using logarithms to solve for the exponent, as logarithms are the inverse operations of exponentiation.

Logarithmic Properties

Logarithmic properties, or identities, are tools that help simplify and solve equations involving logarithms. These identities reflect the underlying relationships between exponential functions and logarithms. For instance, one of the fundamental properties states that the logarithm of an exponential function with the same base, such as ln(e^x), simplifies to just x.

This property is not arbitrary; it makes sense if you consider that taking the logarithm of a number is like asking 'to what exponent do I need to raise the base to obtain this number?' When the base of the exponential matches that of the logarithm—as seen in the exercise—the answer is straightforward: the exponent itself. Other properties of logarithms that students should be familiar with include the product, quotient, and power rules, which respectively allow for the simplification of logarithms of products, quotients, and exponents.