Problem 51 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the expression. $$-1+\ln e^{2 x}$$

Step-by-Step Solution

Answer

-1 + \(2x\)

Step 1: Identify the Natural Logarithm Transformation Identity

Recognize the logarithmic identity of base \(e\), \(\ln e^{x} = x\). This identity comes from the fact that \(e\) and \(\ln\) are inverse operations, therefore \(\ln e^{x}\) simplifies to \(x\).

Step 2: Apply the Identity to Simplify the Expression

Apply the identity to \(\ln e^{2x}\), this simplifies to \(2x\). Substituting this into the original expression, we get -1 + \(2x\).

Key Concepts

Understanding Natural Logarithms

When studying logarithms, one special logarithm that frequently appears is the natural logarithm . The natural logarithm is denoted as ( \( \ln(x) \) ) and is unique in that its base is the mathematical constant ( \( e \approx 2.718 \) ), also known as Euler's number. This particular logarithm is widely used due to its natural occurrence in many areas of mathematics, including calculus and complex analysis.

Natural logarithms are used to solve equations where the variable is in an exponent, and they help simplify growth and decay problems, such as those found in biology and finance. Notably, ( \( \ln(e) \) ) is always 1, because ( \( e \) ) raised to the power of 1 is ( \( e \) ). Similarly, ( \( \ln(1) \) ) is 0, which is consistent with any number raised to the power of 0 being 1. This serves as the foundation for natural logarithmic identities that many students encounter in algebra and pre-calculus courses.

Exploring Logarithmic Identities

Logarithmic identities are formulae that help simplify expressions involving logarithms, making complex calculations more manageable. One fundamental identity involves the logarithm of an exponent, specifically ( \( \ln(a^x) = x \cdot \ln(a) \) ), where ( \( a \) ) is the base of the logarithm. This is particularly useful when dealing with the natural logarithm since the base is Euler's number ( \( e \) ).

Using this identity, one can transform a logarithmic expression with an exponent to a simple algebraic form. For example, ( \( \ln(e^{2x}) = 2x \cdot \ln(e) \) ) can be simplified knowing that ( \( \ln(e) = 1 \) ), hence resulting in just ( \( 2x \) ). It's worth noting that these identities hold for all logarithms regardless of the base, not just natural logarithms. Applying these identities effectively can greatly simplify the process of solving logarithmic equations.

Inverse Operations in Algebra

In mathematics, the concept of inverse operations is used to cancel out functions and solve for unknown variables. Two functions are considered inverses of each other if applying one function followed by the other brings you back to your original value. For instance, addition and subtraction are inverse operations: adding 3 to a number and then subtracting 3 returns you to your starting number.

In the context of logarithms, the exponential function and natural logarithm are inverses. This is why the expression ( \( \ln(e^{x}) \) ) resolves to ( \( x \) ) because the ( \( \ln \) ) function 'undoes' the exponentiation by ( \( e \) ). Understanding this inverse relationship is crucial when learning how to solve equations involving exponential growth or decay, as well as when dealing with interest calculations in finance, where these principles are frequently applied. Recognizing and utilizing inverse operations can simplify complex expressions and are an essential tool in a student's algebra skill set.