Solving Exponential Equations: 2x -4

Understanding how to solve exponential equations is crucial in mathematics. However, the equation 2x -4 presents a unique challenge. Let's explore the steps involved and uncover the complex solutions.

Solving for x When 2x -4

When dealing with an equation like 2x -4, we encounter an interesting scenario. Normally, we would expect a base raised to an exponent to result in a positive value. However, in this case, we are dealing with a negative value, which leads us to investigate complex solutions.

Step-by-Step Solution:

1. Taking the Natural Logarithm

To solve for x, let's take the natural logarithm (ln) of both sides of the equation:

ln(2x) ln(-4)

2. Using Logarithm Properties

Using the logarithm property ln(a^b) b * ln(a), we can simplify the left side of the equation:

x * ln(2) ln(-4)

Note on the Complex Logarithm

The logarithm of a negative number is not defined in the real number system. Therefore, we need to extend our solution to the complex number system. The complex logarithm of a negative real number is given by:

ln(-a) ln(a) iπ 2πi

Thus, when solving for x, we have:

x * ln(2) ln(4) iπ 2πni

3. Simplifying the Equation

Simplifying further, we get:

x (ln(4) iπ 2πni) / ln(2)

For any integer value of n:

x (ln(4) iπ 2πni) / ln(2)

This represents the complex solution for x.

Alternative Approach:

1. Using Euler's Formula

Euler's formula states that:

eipi -1

Therefore, it follows that:

eln(4)ipi -4

We can express 2x in terms of eax:

2x eln(2)x eax

Making the substitution:

eax -4

Using the complex logarithm, we have:

ax ln(4) iπ 2πni

Solving for x:

x (ln(4) iπ 2πni) / ln(2)

Conclusion:

While the equation 2x -4 does not have a real solution, it does have complex solutions. The steps in this solution showcase the use of logarithms and complex numbers.

Key Points:

Complex solutions arise when dealing with negative bases in exponential equations. Euler's formula and logarithmic properties help in solving such equations. The solution involves using the complex logarithm of a negative number.

Understanding these concepts is essential in advanced mathematics and engineering.