$x^2 -2x+65$ and $5-x$ are two algebraic expressions. The value of logarithm of $x^2 -2x+65$ to $5-x$ is $2$. The value of $x$ can be determined by solving this logarithmic equation.

$\log_{5-x} {(x^2 -2x+65)} = 2$

The logarithmic equation can be expressed in exponential form by the relation between logarithms and exponential functions.

$\implies x^2 -2x+65 = {(5-x)}^2$

Expand the right hand side expression by the square of difference of two terms.

$\implies x^2 -2x+65 = 5^2 + x^2 -2 \times 5 \times x$

$\implies x^2 -2x+65 = x^2 -10x +25$

$\implies x^2 -x^2 -2x +10x = -65+25$

$\implies \require{cancel} \cancel{x^2} -\cancel{x^2} +8x = -40$

$\implies 8x = -40$

$\implies x = \dfrac{-40}{8}$

$\implies \require{cancel} x = -\dfrac{\cancel{40}}{\cancel{8}}$

$\therefore \,\,\,\,\,\, x = -5$

Therefore, it is solved that the value of $x$ is $-5$ and it is the required solution for this logarithmic problem in algebraic form.

The value of $x$ can be verified mathematically by substituting the variable $x$ by $-5$ in the left hand side logarithmic expression. If the logarithmic expression is given the right hand side value and it is $2$, then the value of $x = -5$ is true.

$\log_{5-x} {(x^2 -2x+65)}$

Put $x = -5$ and simplify the expression.

$= \log_{(5-(-5)} {[{(-5)}^2 -2(-5)+65]}$

$= \log_{10} {(25+10+65)}$

$= \log 100$

$= \log 10^2$

$= 2$

Therefore, the value of $x$ is $-5$ and it is required solution for this maths problem.

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