What is log_{10}(p^x(1-p)^{n-x}) ?

The solution to log_{10}(p^x(1-p)^{n-x}) is xlog_{10}(p)+(n-x)log_{10}(1-p)

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simplify log_{10}(p^x(1-p)^{n-x})

simplify

lo g_{10} (p^{x} (1 - p)^{n - x})

x lo g_{10} (p) + (n - x) lo g_{10} (1 - p)

Solution steps

lo g_{10} (p^{x} (1 - p)^{n - x})

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

lo g_{10} (p^{x} (1 - p)^{n - x}) = lo g_{10} (p^{x}) + lo g_{10} ((1 - p)^{n - x})

= lo g_{10} (p^{x}) + lo g_{10} ((- p + 1)^{- x + n})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

lo g_{10} (p^{x}) = x lo g_{10} (p)

= x lo g_{10} (p) + lo g_{10} ((1 - p)^{n - x})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

lo g_{10} ((1 - p)^{n - x}) = (n - x) lo g_{10} (1 - p)

= x lo g_{10} (p) + (n - x) lo g_{10} (1 - p)

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What is log_{10}(p^x(1-p)^{n-x}) ?
The solution to log_{10}(p^x(1-p)^{n-x}) is xlog_{10}(p)+(n-x)log_{10}(1-p)