Solving the Equation $|1 - 2i|^x = 5^x$ for Non-Zero Integral Solutions

We are asked to find the number of non-zero integral solutions for the equation $|1 - 2i|^x = 5^x$. Let's break down the problem step by step.

The equation involves the modulus of a complex number and exponents. First, we need to calculate the modulus of the complex number $1 - 2i$.

Calculating the Modulus of a Complex Number

For a complex number $a + bi$, the modulus is given by the formula $|a + bi| = \sqrt{a^2 + b^2}$.

In our complex number $1 - 2i$, we have $a = 1$ and $b = -2$.

So, the modulus $|1 - 2i|$ is:

\(|1 - 2i| = \sqrt{1^2 + (-2)^2}\)

\(|1 - 2i| = \sqrt{1 + 4}\)

\(|1 - 2i| = \sqrt{5}\)

Substituting the Modulus back into the Equation

Now we substitute the calculated modulus back into the original equation $|1 - 2i|^x = 5^x$.

The equation becomes:

\((\sqrt{5})^x = 5^x\)

Solving the Exponential Equation

To solve this equation, we can express $\sqrt{5}$ as a power of 5. We know that $\sqrt{5} = 5^{1/2}$.

Substitute this into the equation:

\((5^{1/2})^x = 5^x\)

Using the exponent rule \((a^m)^n = a^{mn}\), we simplify the left side:

\(5^{(1/2)x} = 5^x\)

Now we have an equation where the bases are the same (both are 5). For the equality to hold, the exponents must be equal.

\(\frac{1}{2}x = x\)

To solve for \(x\), we can rearrange the equation:

\(x - \frac{1}{2}x = 0\)

\(x\left(1 - \frac{1}{2}\right) = 0\)

\(x\left(\frac{1}{2}\right) = 0\)

\(\frac{x}{2} = 0\)

Multiplying both sides by 2 gives:

\(x = 0\)

Identifying Integral Solutions

An integral solution is a solution where \(x\) is an integer. The solution we found is \(x = 0\), which is indeed an integer.

So, the only integral solution to the equation \(|1 - 2i|^x = 5^x\) is \(x = 0\).

Counting Non-Zero Integral Solutions

The question specifically asks for the number of *non-zero* integral solutions. The only integral solution we found is \(x = 0\). The number zero is not a non-zero value.

Since the only integral solution is 0, and 0 is not non-zero, there are no non-zero integral solutions.

Therefore, the number of non-zero integral solutions is zero.

Revision Table: Key Concepts

Additional Information on Exponential Equations and Complex Numbers

Exponential equations often involve variables in the exponent. To solve \(a^x = b\), you might use logarithms. In this problem, we had \(a^x = a^y\) form, which simplifies to \(x=y\) if \(a > 0\) and \(a \neq 1\).

Complex numbers are numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit defined by \(i^2 = -1\). The modulus represents the magnitude or length of the vector corresponding to the complex number in the complex plane.

Understanding the properties of exponents and the definition of complex number modulus is crucial for solving such equations.