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Popular >

simplify 2ln(3)+3ln(27)-2ln(9)

Solution

simplify

2 ln (3) + 3 ln (27) - 2 ln (9)

Solution

7 ln (3)

+1

Decimal Notation

7.69028 \dots

Solution steps

2 ln (3) + 3 ln (27) - 2 ln (9)

Apply log rule:

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

3 ln (27) = ln (2 7^{3})

= 2 ln (3) + ln (2 7^{3}) - 2 ln (9)

Apply log rule:

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

2 ln (9) = ln (9^{2})

= 2 ln (3) + ln (2 7^{3}) - ln (9^{2})

Apply log rule:

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

ln (2 7^{3}) - ln (9^{2}) = ln (\frac{2 7 ^{3}}{9 ^{2}})

= 2 ln (3) + ln (\frac{2 7 ^{3}}{9 ^{2}})

\frac{2 7 ^{3}}{9 ^{2}} = 243

= 2 ln (3) + ln (243)

Apply log rule:

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

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

= ln (3^{2}) + ln (243)

Apply log rule:

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

ln (3^{2}) + ln (243) = ln (3^{2} \cdot 243)

= ln (3^{2} \cdot 243)

3^{2} \cdot 243 = 2187

= ln (2187)

Rewrite

2187

in power-base form:

2187 = 3^{7}

= ln (3^{7})

Apply log rule:

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

ln (3^{7}) = 7 ln (3)

= 7 ln (3)

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Frequently Asked Questions (FAQ)

What is 2ln(3)+3ln(27)-2ln(9) ?
The solution to 2ln(3)+3ln(27)-2ln(9) is 7ln(3)