Let's analyze each point to determine if it lies on the curve defined by the equation [tex]\( y = 5^x \)[/tex]:
1. Point (1, 5):
- Here, [tex]\( x = 1 \)[/tex] and [tex]\( y = 5 \)[/tex].
- Substitute [tex]\( x \)[/tex] into the equation [tex]\( y = 5^x \)[/tex]:
[tex]\[
y = 5^1 = 5.
\][/tex]
- Thus, [tex]\( y = 5 \)[/tex], which matches the y-coordinate of the point (1, 5). Therefore, this point lies on the curve.
2. Point [tex]\(\left(-1, \frac{1}{5}\right)\)[/tex]:
- Here, [tex]\( x = -1 \)[/tex] and [tex]\( y = \frac{1}{5} \)[/tex].
- Substitute [tex]\( x \)[/tex] into the equation [tex]\( y = 5^x \)[/tex]:
[tex]\[
y = 5^{-1} = \frac{1}{5}.
\][/tex]
- Thus, [tex]\( y = \frac{1}{5} \)[/tex], which matches the y-coordinate of the point [tex]\(\left(-1, \frac{1}{5}\right)\)[/tex]. Therefore, this point lies on the curve.
3. Point (5, 25):
- Here, [tex]\( x = 5 \)[/tex] and [tex]\( y = 25 \)[/tex].
- Substitute [tex]\( x \)[/tex] into the equation [tex]\( y = 5^x \)[/tex]:
[tex]\[
y = 5^5 = 3125.
\][/tex]
- Thus, [tex]\( y = 3125 \)[/tex], which does not match the y-coordinate of the point (5, 25). Therefore, this point does not lie on the curve.
In conclusion, the point that does not lie on the curve [tex]\( y = 5^x \)[/tex] is [tex]\((5, 25)\)[/tex].
