What is 2ln(5)+2ln(2) ?

The solution to 2ln(5)+2ln(2) is 2ln(10)

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simplify 2ln(5)+2ln(2)

simplify

2 ln (5) + 2 ln (2)

2 ln (10)

Decimal Notation

4.60517 \dots

Solution steps

2 ln (5) + 2 ln (2)

Apply log rule:

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

2 ln (5) = ln (5^{2})

= ln (5^{2}) + 2 ln (2)

Apply log rule:

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

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

= ln (5^{2}) + ln (2^{2})

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

ln (5^{2}) + ln (2^{2}) = ln (5^{2} \cdot 2^{2})

= ln (5^{2} \cdot 2^{2})

5^{2} \cdot 2^{2} = 100

= ln (100)

Rewrite

100

in power-base form:

100 = 1 0^{2}

= ln (1 0^{2})

Apply log rule:

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

ln (1 0^{2}) = 2 ln (10)

= 2 ln (10)

ln (- e^{- 2} \cdot 0.2)

lo g_{10} (- 6 i)

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

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

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