What is 1/2 log_{10}(x+3)+1/2 log_{10}(2x-1) ?

The solution to 1/2 log_{10}(x+3)+1/2 log_{10}(2x-1) is log_{10}((x+3)^{1/2}(2x-1)^{1/2})

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simplify 1/2 log_{10}(x+3)+1/2 log_{10}(2x-1)

simplify

\frac{1}{2} lo g_{10} (x + 3) + \frac{1}{2} lo g_{10} (2 x - 1)

lo g_{10} ((x + 3)^{\frac{1}{2}} (2 x - 1)^{\frac{1}{2}})

Solution steps

\frac{1}{2} lo g_{10} (x + 3) + \frac{1}{2} lo g_{10} (2 x - 1)

Apply log rule:

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

\frac{1}{2} lo g_{10} (x + 3) = lo g_{10} ((x + 3)^{\frac{1}{2}})

= lo g_{10} ((x + 3)^{\frac{1}{2}}) + \frac{1}{2} lo g_{10} (2 x - 1)

Apply log rule:

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

\frac{1}{2} lo g_{10} (2 x - 1) = lo g_{10} ((2 x - 1)^{\frac{1}{2}})

= lo g_{10} ((x + 3)^{\frac{1}{2}}) + lo g_{10} ((2 x - 1)^{\frac{1}{2}})

Apply log rule:

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

lo g_{10} ((x + 3)^{\frac{1}{2}}) + lo g_{10} ((2 x - 1)^{\frac{1}{2}}) = lo g_{10} ((x + 3)^{\frac{1}{2}} (2 x - 1)^{\frac{1}{2}})

= lo g_{10} ((x + 3)^{\frac{1}{2}} (2 x - 1)^{\frac{1}{2}})

\frac{ln ( \frac{69}{97} )}{2 ln ( 2 )}

\frac{1}{5} lo g_{3} (10)

- \frac{1}{16} lo g_{e} (\frac{1}{16}) - \frac{15}{16} lo g_{e} (\frac{15}{16})

ln (2 \cdot e^{- 10 k})

lo g_{x} (- 2 lo g_{y} (+ lo g_{z} (x)))

What is 1/2 log_{10}(x+3)+1/2 log_{10}(2x-1) ?
The solution to 1/2 log_{10}(x+3)+1/2 log_{10}(2x-1) is log_{10}((x+3)^{1/2}(2x-1)^{1/2})