What is 3ln(5)+1/2 ln(9)-3ln(3)-5ln(25) ?

The solution to 3ln(5)+1/2 ln(9)-3ln(3)-5ln(25) is -ln(703125)

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simplify 3ln(5)+1/2 ln(9)-3ln(3)-5ln(25)

simplify

3 ln (5) + \frac{1}{2} ln (9) - 3 ln (3) - 5 ln (25)

- ln (703125)

Decimal Notation

- 13.46328 \dots

Solution steps

3 ln (5) + \frac{1}{2} ln (9) - 3 ln (3) - 5 ln (25)

Apply log rule:

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

3 ln (5) = ln (5^{3})

= ln (5^{3}) + \frac{1}{2} ln (9) - 3 ln (3) - 5 ln (25)

Apply log rule:

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

\frac{1}{2} ln (9) = ln (9^{\frac{1}{2}})

= ln (5^{3}) + ln (9^{\frac{1}{2}}) - 3 ln (3) - 5 ln (25)

Apply log rule:

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

ln (5^{3}) + ln (9^{\frac{1}{2}}) = ln (5^{3} \cdot 9^{\frac{1}{2}})

= ln (5^{3} \cdot 9^{\frac{1}{2}}) - 3 ln (3) - 5 ln (25)

5^{3} \cdot 9^{\frac{1}{2}} = 375

= ln (375) - 3 ln (3) - 5 ln (25)

Apply log rule:

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

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

= ln (375) - ln (3^{3}) - 5 ln (25)

Apply log rule:

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

ln (375) - ln (3^{3}) = ln (\frac{375}{3 ^{3}})

= ln (\frac{375}{3 ^{3}}) - 5 ln (25)

\frac{375}{3 ^{3}} = \frac{125}{9}

= ln (\frac{125}{9}) - 5 ln (25)

Apply log rule:

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

5 ln (25) = ln (2 5^{5})

= ln (\frac{125}{9}) - ln (2 5^{5})

Apply log rule:

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

ln (\frac{125}{9}) - ln (2 5^{5}) = ln (\frac{\frac{125}{9}}{2 5 ^{5}})

= ln (\frac{\frac{125}{9}}{2 5 ^{5}})

\frac{\frac{125}{9}}{2 5 ^{5}} = \frac{1}{703125}

= ln (\frac{1}{703125})

Apply log rule:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

= - ln (703125)

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What is 3ln(5)+1/2 ln(9)-3ln(3)-5ln(25) ?
The solution to 3ln(5)+1/2 ln(9)-3ln(3)-5ln(25) is -ln(703125)