What is ln(4)-2ln(8) ?

The solution to ln(4)-2ln(8) is -4ln(2)

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simplify ln(4)-2ln(8)

simplify

ln (4) - 2 ln (8)

- 4 ln (2)

Decimal

- 2.77258 \dots

Solution steps

ln (4) - 2 ln (8)

Apply log rule:

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

2 ln (8) = ln (8^{2})

= ln (4) - ln (8^{2})

Apply log rule:

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

ln (4) - ln (8^{2}) = ln (\frac{4}{8 ^{2}})

= ln (\frac{4}{8 ^{2}})

\frac{4}{8 ^{2}} = \frac{1}{16}

= ln (\frac{1}{16})

Apply log rule:

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

= - ln (16)

Rewrite

16

in power-base form:

16 = 2^{4}

= - ln (2^{4})

Apply log rule:

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

ln (2^{4}) = 4 ln (2)

= - 4 ln (2)

ln (\frac{1 ^{2} + 1}{2 ( 1 ) ^{2} + 1})

ln (\frac{400}{300})

- \frac{1}{9} ln (3)

40 e^{9} - 15 ln (19) + 210 + 15 ln (x)

lo g_{10} (x^{4} \frac{7 y ^{2}}{z})