Question

Solve the equation.\(2 e^{2 x}-7=5\)

Short answer

The solution to the equation is \(x = \frac{\ln(6)}{2}\)

Step-by-step solution

Isolate the exponential term

First let's rearrange this equation in such a way that and isolate the term with exponential base on one side. \[2 e^{2 x} = 5 + 7\]

Simplify

By simplifying, obataining the following equation: \[2 e^{2 x} = 12\]

Divide by 2

We can simplify further by dividing through by 2 to isolate \(e^{2x}\) : \[e^{2x} = 6\]

Apply the natural logarithm function

Now apply the natural log function on both sides because It is the inverse of the exponential function: \[\ln(e^{2x}) = \ln(6)\]

Use the property of logarithms that brings down the exponent

Let's simplify our equation using the property of logarithms that allows to bring down the exponent: \[2x = \ln(6)\]

Divide both sides by 2 to solve for x

Finally, to solve for \(x\), divide both sides by 2: \[x = \frac{\ln(6)}{2}\]

Key concepts

Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent, such as \(e^{2x}\). These functions are incredibly important in many areas of science and mathematics due to their consistent rate of growth or decay. In simpler terms, in an exponential function, the rate of increase or decrease is proportional to the current value, leading to rapid changes in values.

• **Base and Exponent**: In an exponential function, a base number (like 2 or \(e\)) is repeatedly multiplied by itself, based on the value of the exponent (\(x\)).
• **Natural Base \(e\)**: The number \(e\) (approximately 2.718) is a special base that's used often in calculus and natural processes because of its unique properties related to continuous growth.

In the exercise, we start with \(2e^{2x} - 7 = 5\). By understanding exponential functions, we know to isolate and manipulate this part of the equation first.

Natural Logarithms

A natural logarithm is simply a logarithm where the base is \(e\), the natural exponential number. It's denoted as \(\ln\). Natural logarithms are used to "undo" exponential functions because they are inverse operations.

• **Inverse Relationship**: The natural log and exponential functions are inverses, meaning \(\ln(e^x) = x\). This relationship is essential for solving equations involving exponentials.
• **Application to Equations**: In our exercise, applying \(\ln\) helps simplify \(e^{2x} = 6\) to \(\ln(e^{2x}) = \ln(6)\), allowing us to tackle the exponent directly.

Using \(\ln\), you can actually transform the problem into a form where you can more easily solve for the unknown variable (\(x\)).

Properties of Logarithms

Logarithms have several useful properties that help simplify and solve equations, especially those involving exponentials. In the context of solving \(e^{2x} = 6\), certain properties are particularly beneficial.

• **Exponent Property**: One key property is \(\ln(a^b) = b\ln(a)\). This is the property used in the exercise to bring down the exponent \(2x\), simplifying \(\ln(e^{2x})\) to \(2x\ln(e)\).
• **Base-\(e\) Simplifications**: Since \(\ln(e) = 1\), many calculations become simpler. For example, \(2x\ln(e) = 2x\times1 = 2x\), as used in solving the equation.

These properties make logarithms a powerful tool for transitions between complex exponential equations and simpler linear equations, which are easier to solve. They help break down the problem into manageable steps, facilitating the solution of equations involving exponentials.