What is log_{10}((1/((1+cot^2(x))^1))/2) ?

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

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simplify log_{10}((1/((1+cot^2(x))^1))/2)

simplify

lo g_{10} (\frac{\frac{1}{( 1 + c o t ^{2} ( x ) ) ^{1}}}{2})

- lo g_{10} (cot^{2} (x) + 1) - lo g_{10} (2)

Solution steps

lo g_{10} (\frac{\frac{1}{( 1 + c o t ^{2} ( x ) ) ^{1}}}{2})

Apply log rule:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

lo g_{10} (\frac{\frac{1}{( 1 + c o t ^{2} ( x ) ) ^{1}}}{2}) = lo g_{10} (\frac{1}{( 1 + cot ^{2} ( x ) ) ^{1}}) - lo g_{10} (2)

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

Apply log rule:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

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

= lo g_{10} (1) - lo g_{10} ((cot^{2} (x) + 1)^{1}) - lo g_{10} (2)

Apply log rule:

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

lo g_{10} ((1 + cot^{2} (x))^{1}) = 1 \cdot lo g_{10} (1 + cot^{2} (x))

= lo g_{10} (1) - 1 \cdot lo g_{10} (1 + cot^{2} (x)) - lo g_{10} (2)

Apply log rule:

lo g_{a} (1) = 0

= 0 - 1 \cdot lo g_{10} (cot^{2} (x) + 1) - lo g_{10} (2)

Refine

= - lo g_{10} (cot^{2} (x) + 1) - lo g_{10} (2)

ln (\frac{1}{40})

ln (\frac{23138}{1500})

2 (ln (\frac{x - 25}{25} + 1) - ln (\frac{x - 25}{25} - 1))

ln (40) - ln (400)

ln (\frac{x - 1}{x}) + c

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