Certainly! To solve the equation [tex]\(4^{x-3}=8\)[/tex], let's work through the problem step-by-step.
1. Understand the Form of the Equation:
The given equation is [tex]\(4^{x-3} = 8\)[/tex].
2. Express Both Sides with the Same Base:
The left side, [tex]\(4^{x-3}\)[/tex], is already in terms of base 4. We should try to express [tex]\(8\)[/tex] with a related base for simpler calculations. Notice that:
[tex]\[
4 = 2^2 \quad \text{and} \quad 8 = 2^3
\][/tex]
Thus, we can rewrite [tex]\(4^{x-3}\)[/tex] using base 2:
[tex]\[
4 = 2^2 \Rightarrow 4^{x-3} = (2^2)^{x-3} = 2^{2(x-3)} = 2^{2x-6}
\][/tex]
So our equation becomes:
[tex]\[
2^{2x-6} = 2^3
\][/tex]
3. Set the Exponents Equal:
Since we now have the same base (2) on both sides of the equation, we can equate the exponents:
[tex]\[
2x - 6 = 3
\][/tex]
4. Solve the Equation for [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], follow these steps:
[tex]\[
2x - 6 = 3
\][/tex]
Add 6 to both sides:
[tex]\[
2x = 9
\][/tex]
Divide by 2:
[tex]\[
x = \frac{9}{2}
\][/tex]
So, one solution to the equation [tex]\(4^{x-3} = 8\)[/tex] is [tex]\( x = \frac{9}{2} \)[/tex].
5. Consider Complex Solutions:
In many exponential equations, there can also be solutions involving complex numbers. By analyzing such solutions, we find another form:
[tex]\[
\left( \frac{\log(512)}{2} + \frac{i \pi}{\log(2)} \right)
\][/tex]
Thus, the complete solutions to the equation are:
[tex]\[
x = \frac{9}{2} \quad \text{and} \quad x = \left( \frac{\log(512)}{2} + \frac{i \pi}{\log(2)} \right)
\][/tex]
Conclusion:
On a graph, the real solution [tex]\(\frac{9}{2}\)[/tex] (which is 4.5) would be shown as the point where the curve of [tex]\(4^{x-3}\)[/tex] intersects the horizontal line [tex]\(y=8\)[/tex]. The complex solution usually is not represented on a simple Cartesian graph but is noted for completeness.
Therefore, the graph that accurately represents the solution will show an intersection at [tex]\( x = \frac{9}{2} \)[/tex].
