First, rewrite the equation [tex]25^{z+2} = 125[/tex] with the base equal to 5. Note that:
- [tex]25=5^2;[/tex]
- [tex]125=5^3;[/tex]
- [tex]25^{z+2}=(5^2)^{z+2}=5^{2\cdot (z+2)}.[/tex]
Then
[tex]5^{2\cdot (z+2)}=5^3.[/tex]
Since the bases are equal, you have that powers are equal too:
[tex]2\cdot (z+2)=3,\\ \\2z+4=3,\\ \\2z=3-4,\\ \\2z=-1,\\ \\z=-\dfrac{1}{2}.[/tex]
Answer: [tex]z=-\dfrac{1}{2}.[/tex]
