What is ln(0.1^2*0.5^2*1^2*2^2*4^2) ?

The solution to ln(0.1^2*0.5^2*1^2*2^2*4^2) is 2ln(0.1)+2ln(0.5)+6ln(2)

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simplify ln(0.1^20.5^21^22^24^2)

simplify

ln (0. 1^{2} \cdot 0. 5^{2} \cdot 1^{2} \cdot 2^{2} \cdot 4^{2})

2 ln (0.1) + 2 ln (0.5) + 6 ln (2)

Decimal Notation

- 1.83258 \dots

Solution steps

ln (0. 1^{2} \cdot 0. 5^{2} \cdot 1^{2} \cdot 2^{2} \cdot 4^{2})

Apply log rule:

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

ln (0. 1^{2} \cdot 0. 5^{2} \cdot 1^{2} \cdot 2^{2} \cdot 4^{2}) = ln (0. 1^{2}) + ln (0. 5^{2}) + ln (1^{2}) + ln (2^{2}) + ln (4^{2})

= ln (0. 1^{2}) + ln (0. 5^{2}) + ln (1^{2}) + ln (2^{2}) + ln (4^{2})

Apply log rule:

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

ln (0. 1^{2}) = 2 ln (0.1)

= 2 ln (0.1) + ln (0. 5^{2}) + ln (1^{2}) + ln (2^{2}) + ln (4^{2})

Apply log rule:

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

ln (0. 5^{2}) = 2 ln (0.5)

= 2 ln (0.1) + 2 ln (0.5) + ln (1^{2}) + ln (2^{2}) + ln (4^{2})

Apply log rule:

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

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

= 2 ln (0.1) + 2 ln (0.5) + 2 ln (1) + ln (2^{2}) + ln (4^{2})

Apply log rule:

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

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

= 2 ln (0.1) + 2 ln (0.5) + 2 ln (1) + 2 ln (2) + ln (4^{2})

Apply log rule:

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

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

= 2 ln (0.1) + 2 ln (0.5) + 2 ln (1) + 2 ln (2) + 2 ln (4)

2 ln (1) = 0

= 2 ln (0.1) + 2 ln (0.5) + 0 + 2 ln (2) + 2 ln (4)

2 ln (0.1) + 2 ln (0.5) + 0 + 2 ln (2) + 2 ln (4) = 2 ln (0.1) + 2 ln (0.5) + 2 ln (2) + 2 ln (4)

= 2 ln (0.1) + 2 ln (0.5) + 2 ln (2) + 2 ln (4)

2 ln (4) = 4 ln (2)

= 2 ln (0.1) + 2 ln (0.5) + 2 ln (2) + 4 ln (2)

Add similar elements:

2 ln (2) + 4 ln (2) = 6 ln (2)

= 2 ln (0.1) + 2 ln (0.5) + 6 ln (2)

3 ln (b) - 3 ln (a)

ln (3 x^{2}) + ln (7 y)

lo g_{6} (\frac{2 x}{y})

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

1.5 - [lo g_{9} (t) + 3 lo g_{9} (p)]

What is ln(0.1^2*0.5^2*1^2*2^2*4^2) ?
The solution to ln(0.1^2*0.5^2*1^2*2^2*4^2) is 2ln(0.1)+2ln(0.5)+6ln(2)