What is expand log_{1/2}((3x^2)/2) ?

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

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expand log_{1/2}((3x^2)/2)

expand

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

lo g_{\frac{1}{a}} (x) = - lo g_{a} (x)

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

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

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

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

= - lo g_{2} (3) - 2 lo g_{2} (x) - (- 1)

Apply rule

- (- a) = a

= - lo g_{2} (3) - 2 lo g_{2} (x) + 1

3 lo g_{10} (x) + 2 lo g_{10} (y) - 5 lo g_{10} (z)

2 ln (3) - ln (2)

2 ln (3) - ln (5)

2 lo g_{10} (3 x) - lo g_{10} (7 x)

5 ln (x) + 3 ln (y) - 2 ln (z)

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