Question

Verifying Properties of logarithms In Exercise,use a graphing utility to verify that the functions are equivalent for x > 0.

f(x) = In (x(x^2 + 1))1/2
g(x) = 1/2[In x + In(x^2 + 1)]

Answer 1

Considering the Product and the Power rule of the Logarithms, they're both equivalent for x >0

Explanation:

1) Considering the Product and the Power rule of the Logarithms:

`\\log_(c)(ab)=log_(c)a+log_(c)b`

`\\log_(c)a^(b)=blog_(c)`

2) Therefore we can say that:

`\\f(x) = ln ((x)(x^2 + 1))^{(1)/(2)}\\\Rightarrow f(x)=ln (x(x^2 + 1))^{(1)/(2)}\Rightarrow f(x)=(1)/(2)ln((x)(x^(2) + 1)).\\g(x) = (1)/(2)(ln (x) + ln(x^(2) + 1))\Rightarrow g(x)=(1)/(2)ln(x(x^2+1))`