What is ln(sqrt(3x))-ln(sqrt(1/2 x)) ?

The solution to ln(sqrt(3x))-ln(sqrt(1/2 x)) is 1/2 ln(6)

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simplify ln(sqrt(3x))-ln(sqrt(1/2 x))

simplify

ln (3 x) - ln (\frac{1}{2} x)

\frac{1}{2} ln (6)

Decimal Notation

0.89587 \dots

Solution steps

ln (3 x) - ln (\frac{1}{2} x)

Apply log rule:

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

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

= ln \frac{3 x}{\frac{1}{2} x}

\frac{3 x}{\frac{1}{2} x} = 6

= ln (6)

Rewrite as

= ln (6^{\frac{1}{2}})

Apply log rule

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

assuming

x \geq 0

= \frac{1}{2} ln (6)

lo g_{b} (\frac{x ^{3} y}{z ^{6}})

lo g_{2} (\frac{( x - 4 ) ^{2}}{x})

3 ln (x) + 3 ln (y)

lo g_{10} (x^{2}) y^{2}

lo g_{16} (\frac{4 x ^{5}}{z})

What is ln(sqrt(3x))-ln(sqrt(1/2 x)) ?
The solution to ln(sqrt(3x))-ln(sqrt(1/2 x)) is 1/2 ln(6)