What is 2ln(40)-2ln(20) ?

The solution to 2ln(40)-2ln(20) is 2ln(2)

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simplify 2ln(40)-2ln(20)

simplify

2 ln (40) - 2 ln (20)

2 ln (2)

Decimal Notation

1.38629 \dots

Solution steps

2 ln (40) - 2 ln (20)

Apply log rule:

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

2 ln (40) = ln (4 0^{2})

= ln (4 0^{2}) - 2 ln (20)

Apply log rule:

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

2 ln (20) = ln (2 0^{2})

= ln (4 0^{2}) - ln (2 0^{2})

Apply log rule:

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

ln (4 0^{2}) - ln (2 0^{2}) = ln (\frac{4 0 ^{2}}{2 0 ^{2}})

= ln (\frac{4 0 ^{2}}{2 0 ^{2}})

\frac{4 0 ^{2}}{2 0 ^{2}} = 4

= ln (4)

Rewrite

4

in power-base form:

4 = 2^{2}

= ln (2^{2})

Apply log rule:

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

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

= 2 ln (2)

\frac{ln ( \frac{5}{2} )}{5 ln ( 10 )}

lo g_{10} (\frac{x ^{2} y ^{2}}{z ^{15}})

3 lo g_{a} (5 x^{2}) - \frac{1}{4} lo g_{a} (8 x + 19)

lo g_{10} ((5 x - 3)^{2}) + lo g_{10} ((2 x + 3)^{3})

[3 ln (3) - 1 - 2 ln (2)] - [3 ln (3) - 1 - 2 ln (2)]