What is 1/2 ln(x-9)+1/2 ln(x) ?

The solution to 1/2 ln(x-9)+1/2 ln(x) is ln((x-9)^{1/2}x^{1/2})

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simplify 1/2 ln(x-9)+1/2 ln(x)

simplify

\frac{1}{2} ln (x - 9) + \frac{1}{2} ln (x)

ln ((x - 9)^{\frac{1}{2}} x^{\frac{1}{2}})

Solution steps

\frac{1}{2} ln (x - 9) + \frac{1}{2} ln (x)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{2} ln (x - 9) = ln ((x - 9)^{\frac{1}{2}})

= ln ((x - 9)^{\frac{1}{2}}) + \frac{1}{2} ln (x)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{2} ln (x) = ln (x^{\frac{1}{2}})

= ln ((x - 9)^{\frac{1}{2}}) + ln (x^{\frac{1}{2}})

Apply log rule:

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

ln ((x - 9)^{\frac{1}{2}}) + ln (x^{\frac{1}{2}}) = ln (x^{\frac{1}{2}} (x - 9)^{\frac{1}{2}})

= ln ((x - 9)^{\frac{1}{2}} x^{\frac{1}{2}})

lo g_{14} (\frac{14}{y})

lo g_{10} (\frac{1}{120})

lo g_{10} (\frac{y ^{5}}{x})

\frac{3}{8} \cdot ln (\frac{8}{3})

ln (\frac{a \cdot b}{c})

What is 1/2 ln(x-9)+1/2 ln(x) ?
The solution to 1/2 ln(x-9)+1/2 ln(x) is ln((x-9)^{1/2}x^{1/2})