How do you expand $ln (\frac{5 \sqrt{y}}{x^{2}})$ $ln ((5sqrty)/x^2)$ ?

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Answer 1 · 2018-04-10T09:47:15

$\frac{1}{2} ln 5 + \frac{1}{2} ln y - 2 ln x$ $color(blue)(1/2ln5+1/2lny-2lnx)$

The law of logarithms state:

${log}_{c} (\frac{a}{b}) = {log}_{c} a - {log}_{c} b$ $log_c(a/b)=log_ca-log_cb$

${log}_{c} (a b) = {log}_{c} a + {log}_{c} b$ $log_c(ab)=log_ca+log_cb$

${log}_{c} a^{b} = b {log}_{c} a$ $log_ca^b=blog_ca$

Notice we can write:

$5 \sqrt{y} = 5 y^{\frac{1}{2}}$ $5sqrt(y)=5y^(1/2)$

$:.$

$ln((5sqrt(y))/x^2)=ln((5(y)^(1/2))/x^2)$

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \=ln5(y)^(1/2)-lnx^2$

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \=1/2ln5y-2lnx$

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \=color(blue)(1/2ln5+1/2lny-2lnx)$

How do you expand ln ((5sqrty)/x^2)ln(5√yx2)ln ((5sqrty)/x^2)?