What is 1/3*ln(64)-ln(12) ?

The solution to 1/3*ln(64)-ln(12) is -ln(3)

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simplify 1/3*ln(64)-ln(12)

simplify

\frac{1}{3} \cdot ln (64) - ln (12)

- ln (3)

Decimal Notation

- 1.09861 \dots

Solution steps

\frac{1}{3} ln (64) - ln (12)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{3} ln (64) = ln (6 4^{\frac{1}{3}})

= ln (6 4^{\frac{1}{3}}) - ln (12)

Apply log rule:

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

ln (6 4^{\frac{1}{3}}) - ln (12) = ln (\frac{6 4 ^{\frac{1}{3}}}{12})

= ln (\frac{6 4 ^{\frac{1}{3}}}{12})

\frac{6 4 ^{\frac{1}{3}}}{12} = \frac{1}{3}

= ln (\frac{1}{3})

Apply log rule:

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

= - ln (3)

5 ln (x - 6) - ln (x^{2} - 11 x + 30) + 2 ln (x - 5)

5 ln (x + 1) + 6 ln (x)

\frac{1}{2} lo g_{10} (b) + 5 lo g_{10} (c)

ln (\frac{81162081}{390625})

lo g_{10} (\frac{6 x}{x + 2})