What is (2ln(3)-2ln(9))-(2ln(2)-2ln(7)) ?

The solution to (2ln(3)-2ln(9))-(2ln(2)-2ln(7)) is ln(49/36)

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simplify (2ln(3)-2ln(9))-(2ln(2)-2ln(7))

simplify

(2 ln (3) - 2 ln (9)) - (2 ln (2) - 2 ln (7))

ln (\frac{49}{36})

Decimal Notation

0.30830 \dots

Solution steps

(2 ln (3) - 2 ln (9)) - (2 ln (2) - 2 ln (7))

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

\frac{2 ^{2}}{7 ^{2}} = \frac{4}{49}

= 2 ln (3) - 2 ln (9) - ln (\frac{4}{49})

Apply log rule:

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

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

= 2 ln (3) - ln (9^{2}) - ln (\frac{4}{49})

Rewrite

- ln (9^{2}) - ln (\frac{4}{49})

- (ln (9^{2}) + ln (\frac{4}{49}))

= 2 ln (3) - (ln (9^{2}) + ln (\frac{4}{49}))

Apply log rule:

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

ln (9^{2}) + ln (\frac{4}{49}) = ln (9^{2} \cdot \frac{4}{49})

= 2 ln (3) - ln (9^{2} \cdot \frac{4}{49})

9^{2} \cdot \frac{4}{49} = \frac{324}{49}

= 2 ln (3) - ln (\frac{324}{49})

Apply log rule:

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

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

= ln (3^{2}) - ln (\frac{324}{49})

Apply log rule:

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

ln (3^{2}) - ln (\frac{324}{49}) = ln (\frac{3 ^{2}}{\frac{324}{49}})

= ln (\frac{3 ^{2}}{\frac{324}{49}})

\frac{3 ^{2}}{\frac{324}{49}} = \frac{49}{36}

= ln (\frac{49}{36})

lo g_{6} (6 y w z)

ln (2) + ln (4 x - 1)

- ln (x + 1) + ln (x - 1)

ln (x - 3) - ln (- 3 + x)

lo g_{2} (\frac{x ^{9}}{5 y})

What is (2ln(3)-2ln(9))-(2ln(2)-2ln(7)) ?
The solution to (2ln(3)-2ln(9))-(2ln(2)-2ln(7)) is ln(49/36)