To solve the equation [tex]\(2^{4x - 2} - 2^{4x - 4} = 2^7 \times 3\)[/tex], let's work through it step by step.
1. Rewrite the equation for clarity:
[tex]\[
2^{4x - 2} - 2^{4x - 4} = 2^7 \times 3
\][/tex]
2. Simplify the terms involving the exponents:
We can rewrite [tex]\(2^{4x - 2}\)[/tex] and [tex]\(2^{4x - 4}\)[/tex] by factoring out the common term. Notice that:
[tex]\[
2^{4x - 2} = 4x - 4 + 2
\][/tex]
Now we factor out [tex]\(2^{4x - 4}\)[/tex]:
[tex]\[
2^{4x - 2} = 2^2 \cdot 2^{4x - 4} = 4 \cdot 2^{4 x-4}
\][/tex]
Therefore, the equation can be simplified to:
[tex]\[
4 \cdot 2^{4 x-4} - 2^{4 x-4} = 2^7 \times 3
\][/tex]
3. Combine like terms:
[tex]\[
(4 -1 ) \cdot 2^{4 x-4} = 2^7 \times 3
\][/tex]
Simplify:
[tex]\[
3 \cdot 2^{4x - 4} = 2^7 \times 3
\][/tex]
4. Divide both sides by 3 to isolate the exponential expression:
[tex]\[
2^{4 x-4 } = 2^7
\][/tex]
5. Since the bases are the same (base 2), we can set the exponents equal to each other:
[tex]\[
4 x-4 = 7
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[
4 x -4 = 7\\
4x = 11\\
x = \frac{11}{4}
\][/tex]
Now, we should consider possible complex solutions due to the periodicity of the exponential function.
Let’s recall the general solution for equations involving periodicity in the context of complex logarithms. When solving [tex]\(a^{bx} = c\)[/tex], given a complex logarithm:
[tex]\[x = \frac{\log(a^c) + 2k \pi i}{b \log(a)}\][/tex]
Transforming our solution back to complex form, we obtain:
[tex]\[
x = \frac{\log_2(2048) + 2k\pi i}{4 \cdot \log_2(2)} = \frac{\log_2(2048) + 2k\pi i}{4}
\][/tex]
After simplifying, the series of solutions are:
[tex]\[
x = \frac{11}{4}
\][/tex]
[tex]\[
x = \frac{\log(2048)/4 + i\pi}{\log(2)}
\][/tex]
[tex]\[
x = \frac{\log(2048) - 2i\pi}{4 \cdot \log(2)}
\][/tex]
[tex]\[
x = \frac{\log(2048) + 2i\pi}{4 \log(2)}
\][/tex]
Thus, the solutions for the equation [tex]\(2^{4x - 2} - 2^{4x - 4} = 2^7 \times 3\)[/tex] are:
[tex]\[
x = \frac{11}{4}, \quad \left(\frac{\log(2048)/4 + i\pi}{\log(2)}, \quad \frac{\log(2048) - 2i\pi}{4\log(2)}, \quad \frac{\log(2048) + 2i\pi}{4\log(2)}\right)
\][/tex]
This gives us both the real and complex solutions to the equation.
