Question

Apply the inverse properties of \(\ln x\) and \(e^{x}\) to simplify the given expression. $$ \ln e^{x^{2}} $$

Short answer

The simplified form of the given expression \(\ln e^{x^{2}}\) is \(x^{2}\).

Step-by-step solution

Apply the Inverse Property of Natural Logarithm and Exponential Function

The inverse property of the natural logarithm and the exponential function states that the natural logarithm of \(e^{x}\) is \(x\). Here, we apply this property to the given expression \(\ln e^{x^{2}}\). We can simply apply the inverse property and remove the \(\ln\)-term and \(e\) - term, leaving behind only the exponent \(x^{2}\).

Key concepts

Natural Logarithm

The natural logarithm, commonly denoted as \(\ln x\), is a mathematical operation that represents the inverse of the exponential function when the base is Euler's number, e, which is approximately 2.71828. This log function is used to determine the power to which e must be raised to get the number x. For example, if we have \(e^y = x\), then \(\ln x = y\).

When working with logarithms, it's important to remember that \(\ln e = 1\), because e raised to the power of 1 is just e. Similarly, \(\ln 1 = 0\) because any number raised to the power of 0 is 1, and e is not an exception. This is especially helpful when simplifying expressions where the natural logarithm and the exponential function cancel each other out due to their inverse relationship.

Exponential Function

The exponential function is a mathematical expression where a number, typically e, is raised to a power. This function is written as \(e^x\), and it represents growth or decay processes such as population growth, radioactive decay, or continuously compounded interest.

The exponential function is unique because it is its own derivative and integral, making it a central concept in calculus. It has the property that the rate of change of the function at any point is directly proportional to the value of the function at that point. This makes the graph of the exponential function a smooth curve that either increases or decreases exponentially, without ever touching the x-axis.

Simplifying Exponential Expressions

Simplifying exponential expressions involves the use of various properties of exponents, such as the power of a power rule, the product of powers rule, and the aforementioned inverse relationship between logarithms and exponentials. When confronted with an expression like \(\ln e^{x^2}\), we can simplify it by recognizing that the natural logarithm and the base-e exponential are inverse operations.

In this instance, they cancel each other out, leaving the exponent \(x^2\) alone. Understanding these properties allows for the simplification of seemingly complex expressions down to much simpler forms, which is a fundamental skill in algebra and calculus.

Calculus

Calculus is a branch of mathematics that studies how quantities change and accumulate. It consists of two main areas: differential calculus, which concerns the rate of change of quantities, and integral calculus, which deals with the accumulation of quantities and areas under curves. Central to calculus are derivatives and integrals—tools that allow us to analyze the behaviors of functions.

When working with functions like the exponential function, calculus provides methods to calculate slopes of tangent lines or the area under the exponential curve. These calculations are essential for solving real-world problems involving rates of change and total accumulated value. Additionally, the inverse relationship between exponential functions and logarithms plays a significant role in solving calculus problems, such as those involving growth and decay models.