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Why is $3 \cdot ln (x) = ln (x^{3})$ $3*ln(x) = ln(x^3)$ ?

1 Answer

Oct 22, 2015

Use the definition of logarithm and basic properties of exponents.
(details below)

Explanation:

Basic definition:
$XXX ln (a) = b$ $color(white)("XXX")ln(a)=b$ means $e^{b} = a$ $e^b=a$

Let $s = 3 ln (x)$ $s =3 ln(x)$
$XXX \Rightarrow ln (x) = \frac{s}{3}$ $color(white)("XXX")rArr ln(x)= s/3$

$XXX \Rightarrow e^{\frac{s}{3}} = x$ $color(white)("XXX")rArr e^(s/3) = x$

$XXX \Rightarrow \sqrt[3]{e^{s}} = x$ $color(white)("XXX")rArr root(3)(e^s) = x$

$XXX \Rightarrow e^{s} = x^{3}$ $color(white)("XXX")rArr e^s = x^3$

$XXX \Rightarrow ln (x^{3}) = s$ $color(white)("XXX")rArrln(x^3) = s$

$XXX \Rightarrow ln (x^{3}) = 3 ln (x)$ $color(white)("XXX")rArrln(x^3) = 3ln(x)$

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