which graph shows the solution to the equation 4^{x - 3}=8?

Explanation:

Step1: Rewrite bases

Rewrite $4$ as $2^2$ and $8$ as $2^3$. So the equation $4^{x - 3}=8$ becomes $(2^2)^{x - 3}=2^3$.

Step2: Apply power - of - a - power rule

By the power - of - a - power rule $(a^m)^n=a^{mn}$, we have $2^{2(x - 3)}=2^3$.

Step3: Set exponents equal

Since the bases are the same, we can set the exponents equal: $2(x - 3)=3$.

Step4: Expand and solve for x

Expand the left - hand side: $2x-6 = 3$. Add $6$ to both sides: $2x=3 + 6=9$. Then divide by $2$: $x=\frac{9}{2}=4.5$. We are looking for a graph of an exponential function $y = 4^{x-3}$ and we want to find the $x$ - value where $y = 8$. The exponential function $y = 4^{x-3}$ is an increasing function. When $x = 4.5$, $y = 8$. We need to find the graph of $y = 4^{x - 3}$ and check which one has the point $(4.5,8)$ on it.

Answer:

We need to analyze the graphs to find the one where the curve of $y = 4^{x-3}$ intersects the horizontal line $y = 8$ at $x = 4.5$. Without seeing the specific details of each graph option, we can't directly pick the correct graph from the list provided, but the above steps show how to find the solution value of $x$ for the given exponential equation. If we assume we are looking at the general shape of the exponential function $y = 4^{x-3}$ (an increasing exponential function shifted 3 units to the right compared to $y = 4^x$) and we know the $x$ - value of the solution is $4.5$, we would look for the graph that has the function value of $8$ at $x = 4.5$.