Solving Exponential Equations And Radians Expressed With Pi

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This article delves into solving exponential equations and expressing angles in radians in terms of π. We will tackle various exponential equations, finding exact solutions for x and explaining the methodologies employed. Furthermore, we will discuss the importance of radians in mathematics and how to convert between degrees and radians, focusing on expressing radian measures as multiples of π. This comprehensive guide aims to enhance your understanding of these fundamental mathematical concepts.

Solving Exponential Equations

In this section, we will address the core task of solving exponential equations. Exponential equations are equations in which the variable appears in the exponent. These types of equations often require specific techniques to isolate the variable and find the exact solution. We will cover two primary types of exponential equations: those that can be solved by isolating the exponential term and using logarithms, and those that require factoring techniques to arrive at the solution. Understanding these methods is crucial for tackling a wide range of mathematical problems in calculus, physics, and engineering. We will proceed step by step, demonstrating each method with detailed explanations and practical examples to ensure clarity and comprehension. The goal is to equip you with the necessary skills to confidently solve exponential equations and apply them in various contexts.

1. $1.15 e^{3 x}-4=6$

To solve the exponential equation $1.15 e^{3 x}-4=6$, our primary goal is to isolate the exponential term. This involves a series of algebraic manipulations to get the term $e^{3x}$ by itself on one side of the equation. Once isolated, we can then apply logarithms to solve for x. Understanding this process is crucial for handling more complex exponential equations. The process allows us to "undo" the exponential function and bring the variable down from the exponent, making it solvable using basic algebraic techniques. This method is a cornerstone for solving many types of exponential problems, making it an essential skill in mathematics and related fields.

First, we add 4 to both sides of the equation:

1.15e3x=101. 15 e^{3 x} = 10

Next, we divide both sides by 1.15:

e3x=101.15=20023e^{3 x} = \frac{10}{1.15} = \frac{200}{23}

Now, to get rid of the exponential, we take the natural logarithm (ln) of both sides:

ln(e3x)=ln(20023)ln(e^{3 x}) = ln(\frac{200}{23})

Using the property that $ln(e^a) = a$, we have:

3x=ln(20023)3x = ln(\frac{200}{23})

Finally, we divide by 3 to solve for x:

x=13ln(20023)x = \frac{1}{3} ln(\frac{200}{23})

This is the exact solution for x. We can approximate this value using a calculator, but the exact solution provides a more precise representation of the answer. Understanding how to manipulate logarithmic expressions and apply logarithmic properties is essential for solving exponential equations effectively. This process not only provides the solution but also deepens the understanding of the relationship between exponential and logarithmic functions, which is fundamental in advanced mathematics.

2. $1.2 e^{2 x}-e^x-110=0$

Solving the exponential equation $1.2 e^{2 x}-e^x-110=0$ requires a different approach. This equation is quadratic in form, meaning we can rewrite it as a quadratic equation by making a suitable substitution. This technique allows us to apply familiar methods for solving quadratic equations, such as factoring or using the quadratic formula. The ability to recognize and manipulate equations into quadratic forms is a powerful skill in algebra and calculus. It allows us to transform seemingly complex equations into simpler, solvable forms. By understanding and applying this method, we can effectively solve a wider range of exponential equations.

Let $y = e^x$. Then, $y^2 = e^{2x}$. Substituting these into the equation, we get:

1.2y2−y−110=01. 2 y^2 - y - 110 = 0

This is a quadratic equation in y. We can solve this by factoring, if possible, or by using the quadratic formula.

Let's try factoring. We are looking for two numbers that multiply to $1.2 imes -110 = -132$ and add up to -1. This might be challenging to factor directly, so let's use the quadratic formula:

y=−b±b2−4ac2ay = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where a = 1.2, b = -1, and c = -110.

y=1±(−1)2−4(1.2)(−110)2(1.2)y = \frac{1 \pm \sqrt{(-1)^2 - 4(1.2)(-110)}}{2(1.2)}

y=1±1+5282.4y = \frac{1 \pm \sqrt{1 + 528}}{2.4}

y=1±5292.4y = \frac{1 \pm \sqrt{529}}{2.4}

y=1±232.4y = \frac{1 \pm 23}{2.4}

So, we have two possible values for y:

y1=1+232.4=242.4=10y_1 = \frac{1 + 23}{2.4} = \frac{24}{2.4} = 10

y2=1−232.4=−222.4=−556y_2 = \frac{1 - 23}{2.4} = \frac{-22}{2.4} = -\frac{55}{6}

Now, we substitute back $e^x$ for y:

  1. ex=10e^x = 10

Taking the natural logarithm of both sides:

x=ln(10)x = ln(10)

  1. ex=−556e^x = -\frac{55}{6}

The exponential function $e^x$ is always positive, so there is no solution for this case.

Thus, the only solution for x is:

x=ln(10)x = ln(10)

This solution demonstrates the importance of recognizing quadratic forms in exponential equations and utilizing appropriate methods like the quadratic formula to solve them. It also highlights the necessity of checking the validity of solutions in the original context of the problem, as some solutions may be extraneous due to the nature of exponential functions.

3.

There appears to be a third question listed as "1.3", but the actual equation is missing. Without the equation, it is impossible to provide a solution or any meaningful discussion. However, we can discuss the general approach to solving exponential equations, which might be relevant once the equation is provided. Typically, solving exponential equations involves isolating the exponential term and then applying logarithms to both sides. If the equation is more complex, it might require substitution or other algebraic techniques to simplify it into a solvable form. When solving exponential equations, it is crucial to double-check the solutions to ensure they are valid and make sense within the context of the original equation. This often involves substituting the solutions back into the original equation to verify they hold true. In summary, without the specific equation, it's challenging to provide a step-by-step solution, but understanding the general methods and precautions will be beneficial when the equation is available.

Expressing Angles in Radians in Terms of π

In mathematics, radians are a crucial unit for measuring angles, especially in calculus and higher-level mathematics. Radians provide a natural and elegant way to express angles, making many formulas simpler and more intuitive. One radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. This definition connects the angle measure directly to the properties of the circle, making radians a fundamental concept in trigonometry and calculus. Understanding radians is essential for working with trigonometric functions, circular motion, and various other mathematical and physical applications. In this section, we will explore how to convert between degrees and radians and, more importantly, how to express radian measures as multiples of π, which is a common and useful practice in many mathematical contexts. We will also discuss why radians are preferred over degrees in many mathematical applications.

Converting Degrees to Radians

The conversion between degrees and radians is based on the fundamental relationship that 180 degrees is equivalent to π radians. This relationship stems from the fact that a semicircle, which spans 180 degrees, has an arc length of π times the radius of the circle. Understanding this equivalence allows us to convert any angle measure from degrees to radians and vice versa. The conversion factor derived from this relationship is π/180 for converting degrees to radians, and 180/π for converting radians to degrees. This conversion is a crucial skill in trigonometry, calculus, and physics, where angles are often expressed in radians due to their mathematical convenience. Mastering this conversion ensures a seamless transition between different units of angle measurement and is essential for solving a wide range of problems involving angles and circular functions.

To convert an angle from degrees to radians, we use the following formula:

Radians=Degrees×π180Radians = Degrees × \frac{\pi}{180}

For example, let's convert 60 degrees to radians:

Radians=60×π180=60π180=π3Radians = 60 × \frac{\pi}{180} = \frac{60\pi}{180} = \frac{\pi}{3}

So, 60 degrees is equal to $\frac{\pi}{3}$ radians.

Common Angles in Radians

Some common angles are frequently used in trigonometry and calculus, and it's helpful to know their radian equivalents in terms of π. Knowing these common angles can significantly speed up calculations and enhance understanding of trigonometric functions. These angles include 0, 30, 45, 60, and 90 degrees, which correspond to 0, π/6, π/4, π/3, and π/2 radians, respectively. Memorizing these conversions and understanding their relationships on the unit circle is invaluable for solving trigonometric equations and understanding the behavior of trigonometric functions. Familiarity with these common radian measures is not only beneficial for academic purposes but also has practical applications in fields such as physics, engineering, and computer graphics, where angles are frequently used in calculations and modeling.

Here are some common angles and their radian equivalents:

  • 0 degrees = 0 radians
  • 30 degrees = $\frac{\pi}{6}$ radians
  • 45 degrees = $\frac{\pi}{4}$ radians
  • 60 degrees = $\frac{\pi}{3}$ radians
  • 90 degrees = $\frac{\pi}{2}$ radians
  • 180 degrees = $\pi$ radians
  • 270 degrees = $\frac{3\pi}{2}$ radians
  • 360 degrees = $2\pi$ radians

Conclusion

In summary, solving exponential equations involves isolating the exponential term and using logarithms, or employing substitution to form a quadratic equation. The process requires a solid understanding of algebraic manipulation and logarithmic properties. Furthermore, expressing angles in radians, particularly in terms of π, is essential for simplifying many mathematical problems, especially in trigonometry and calculus. Radians provide a natural connection between angles and the unit circle, making them a preferred unit of angle measurement in advanced mathematics. Mastering these concepts not only enhances problem-solving skills but also provides a deeper understanding of the underlying principles of mathematics. From isolating exponential terms to applying the quadratic formula and converting between degrees and radians, each step builds upon fundamental mathematical knowledge, fostering a comprehensive grasp of exponential equations and angle measures.