Question

Solve the equation.\(3 e^{4 x}+9=15\)

Short answer

The solution to the equation is \(x = \frac{ln(2)}{4}\)

Step-by-step solution

Remove any constant on the side of the variable

Subtract 9 from both sides of the equation, to have: \(3 e^{4 x} = 15 - 9 = 6\)

Solve for exponential expression

Divide both sides of the equation by 3 to get: \(e^{4 x} = \frac{6}{3} = 2\)

Remove the exponential using natural logarithm

Apply the natural logarithm (ln) on both sides of the equation, to get: \(4x = ln(2)\)

Solve for x

Divide both sides of the equation by 4 to isolate x: \(x = \frac{ln(2)}{4}\)

Key concepts

natural logarithm

The natural logarithm, often abbreviated as "ln," is a logarithm with base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. Thus, the natural logarithm of a number \(a\), expressed as \(\ln(a)\), is the power to which \(e\) must be raised to obtain \(a\).

In the exercise given, the equation involves an exponential term \(e^{4x}\). To solve for \(x\), it's often necessary to "undo" the exponential function, which can be accomplished using the natural logarithm. This is because the natural logarithm is the inverse function of the exponential, allowing us to isolate the variable within the exponent.

When applying the natural logarithm to an equation like \(e^{4x} = 2\), we utilize the property \(\ln(e^a) = a\). Therefore, by taking the natural logarithm of both sides, we simplify our equation, resulting in \(4x = \ln(2)\). This transformation is crucial for moving beyond exponential expressions and solving for the variable.

step-by-step solutions

Step-by-step solutions are a structured approach to solving problems systematically, making it easier to follow and understand each part of the solution process. They are especially valuable when solving complex mathematical equations, as they break down the process into smaller, manageable parts.

In this problem, each step builds upon the previous, ensuring clarity and maintaining the logical flow necessary for solving the equation. It begins by simplifying the original equation \(3 e^{4x} + 9 = 15\). We first subtract the 9 to isolate the exponential term, which is then simplified further by dividing by 3. This reduces the equation to \(e^{4x} = 2\), setting the stage for the use of the natural logarithm.

Each subsequent step methodically progresses: applying the natural logarithm to both sides to transform the equation into a linear form, and finally, solving for \(x\) by dividing both sides by 4. These steps not only help to find the solution but also enhance the students' understanding of each action taken and the reasoning behind it.

mathematical equations

Mathematical equations are expressions that assert the equality of two quantities. Solving equations involves finding the values of variables that make the equation true. There are various types of equations, and this exercise focuses on an exponential equation.

Exponential equations are those in which variables appear in the exponent. The exercise begins with the equation \(3 e^{4x} + 9 = 15\), where \(e^{4x}\) is the exponential term. Such equations are solved by isolating the exponential expression before using logarithms to simplify.

Understanding the structure of mathematical equations helps in determining the steps required to solve them. Key actions include:

• Identifying terms and expressions to simplify.
• Applying mathematical operations to isolate the variable.
• Utilizing logarithmic properties to convert exponential expressions.

These principles are fundamental in approaching and deciphering various mathematical problems, providing a framework that extends beyond any single equation type.