What is ln(18)+ln(4)-2ln(3) ?

The solution to ln(18)+ln(4)-2ln(3) is 3ln(2)

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simplify ln(18)+ln(4)-2ln(3)

simplify

ln (18) + ln (4) - 2 ln (3)

3 ln (2)

Decimal Notation

2.07944 \dots

Solution steps

ln (18) + ln (4) - 2 ln (3)

Apply log rule:

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

ln (18) + ln (4) = ln (18 \cdot 4)

= ln (18 \cdot 4) - 2 ln (3)

Multiply the numbers:

18 \cdot 4 = 72

= ln (72) - 2 ln (3)

Apply log rule:

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

2 ln (3) = ln (3^{2})

= ln (72) - ln (3^{2})

Apply log rule:

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

ln (72) - ln (3^{2}) = ln (\frac{72}{3 ^{2}})

= ln (\frac{72}{3 ^{2}})

\frac{72}{3 ^{2}} = 8

= ln (8)

Rewrite

8

in power-base form:

8 = 2^{3}

= ln (2^{3})

Apply log rule:

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

ln (2^{3}) = 3 ln (2)

= 3 ln (2)

2 ln (c) - 3 ln (d) + ln (b)

ln (\frac{306.34}{1173.15})

e^{2 (l n (x) - l n (1 - x))}

ln (\frac{86 - 60}{86 - 50})

ln (x^{4} (1 - x)^{14})