Answer
$x=4,$ or $ x=-25.$
Work Step by Step
For $x\gt 0$ and $a$ a positive constant other than 1,
$\log_{a}x$ is the exponent to which $a$ must be raised in order to get $x$.
Thus, $\log_{a}x=m$ means $a^{m}=x$
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By definition, $\log_{10}(x^{2}+21x)=2$ means
$10^{2}=x^{2}+21x$
(the solutions must satisfy $x^{2}+21x\gt 0, \qquad(*)$
otherwise the initial equation is not defined)
$100=x^{2}+21x$
$x^{2}+21x-100=0$
$(x-4)(x+25)=0$
Possible solution $x=4$ satisfies the condition (*), so it is a solution.
$16+24(4)\gt 0.$
Possible solution $x=-25$ also satisfies the condition (*)
$625-21(25)\gt 0.$
$x=4,$ or $ x=-25.$
