Question

Solve for \(x\) accurate to three decimal places. (a) \(e^{\ln x}=4\) (b) \(\ln e^{2 x}=3\)

Short answer

The solution to (a) is \(x = 4.000\) and to (b) is \(x = 1.500\).

Step-by-step solution

Solve first equation

We have \(e^{\ln x}=4\). Since \(e^{\ln x}\) is simply \(x\), the equation can be rewritten as \(x = 4\).

Solve second equation

We have \(\ln e^{2 x}=3\). Again, using the property \(\ln(e^x)=x\), we can simplify it to \(2x = 3\). Now, solving for \(x\) gives \(x = \frac{3}{2} = 1.500\) when rounded to three decimal places.

Key concepts

Natural Logarithm

The natural logarithm is a mathematical function that is commonly denoted by \(\ln\). It is the inverse operation of raising \(e\), the irrational and transcendental number approximately equal to 2.71828, to a power. In simpler terms, if we have an equation \(e^y = x\), then the natural logarithm of \(x\) is \(y\), written as \(\ln x = y\).

When we encounter an expression like \(e^{\ln x}\), we are dealing with a fundamental property of logarithms: the base \(e\) raised to the power of its own logarithm will yield the number itself, which is \(x\) in this case. Therefore, \(e^{\ln x} = x\). This principle is central to understanding how to work with exponential equations involving natural logarithms. For instance, in the exercise provided, \(e^{\ln x} = 4\) simplifies directly to \(x = 4\) because of this property.

In practice, the natural logarithm is invaluable for solving equations where the variable is an exponent, which we see in various fields like compound interest in finance, population growth in biology, and decay rates in physics.

Exponential Functions

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. The most common base for an exponential function in higher mathematics is \(e\), leading to functions of the form \(e^x\). These functions exhibit rapid growth or decay and are modeled with the equation \(y = e^x\).

Exponential functions are distinct in their properties, one of which is that the rate of change of the function at any point is proportional to the value of the function at that point. This can be interpreted as the function growing faster as it becomes larger, a key concept in understanding natural phenomena such as populations or investments.

In the context of solving equations, knowing how to manipulate exponential functions is crucial. For example, in the second part of the exercise, \(\ln e^{2x} = 3\), recognizing that the natural logarithm and the exponential function are inverse operations simplifies the equation to \(2x = 3\), which can then be easily solved for \(x\).

Equation Solving

The process of equation solving is a systematic approach to finding the values of variables that make an equation true. It is a fundamental skill in algebra and a basis for higher mathematics and various applications in science and engineering.

The key to effectively solving equations is to understand the operations and relationships involved and to methodically isolate the variable in question. Sometimes this involves applying inverse operations, such as using the natural logarithm to undo an exponentiation, or factoring to break down a polynomial equation.

In the provided exercise, equation solving techniques were utilized to simplify and find the values of \(x\). The first step involved recognizing properties of logarithms and exponential functions, which led to a straightforward solution. The second step used the inverse relationship between natural logarithms and exponents to simplify the equation, allowing \(x\) to be isolated and its value determined. As shown, understanding the underlying concepts and operations is critical for successful equation solving.