en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

simplify 1/2 ln(9)-3ln(6)+3ln(2)

Solution

simplify

\frac{1}{2} ln (9) - 3 ln (6) + 3 ln (2)

Solution

- 2 ln (3)

+1

Decimal Notation

- 2.19722 \dots

Solution steps

\frac{1}{2} ln (9) - 3 ln (6) + 3 ln (2)

Apply log rule:

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

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

= ln (9^{\frac{1}{2}}) - 3 ln (6) + 3 ln (2)

Apply log rule:

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

3 ln (6) = ln (6^{3})

= ln (9^{\frac{1}{2}}) - ln (6^{3}) + 3 ln (2)

Apply log rule:

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

ln (9^{\frac{1}{2}}) - ln (6^{3}) = ln (\frac{9 ^{\frac{1}{2}}}{6 ^{3}})

= ln (\frac{9 ^{\frac{1}{2}}}{6 ^{3}}) + 3 ln (2)

\frac{9 ^{\frac{1}{2}}}{6 ^{3}} = \frac{1}{72}

= ln (\frac{1}{72}) + 3 ln (2)

Apply log rule:

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

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

= ln (\frac{1}{72}) + ln (2^{3})

Apply log rule:

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

ln (\frac{1}{72}) + ln (2^{3}) = ln (2^{3} \cdot \frac{1}{72})

= ln (2^{3} \cdot \frac{1}{72})

\frac{1}{72} \cdot 2^{3} = \frac{1}{9}

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

Apply log rule:

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

= - ln (9)

Rewrite

9

in power-base form:

9 = 3^{2}

= - ln (3^{2})

Apply log rule:

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

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

= - 2 ln (3)

Popular Examples

simplify (ln(13)-3ln(6))/(5ln(6))simplify

\frac{ln ( 13 ) - 3 ln ( 6 )}{5 ln ( 6 )}

simplify 6log_{b}(x+4)-7log_{b}(x+3)+log_{b}(x)simplify

6 lo g_{b} (x + 4) - 7 lo g_{b} (x + 3) + lo g_{b} (x)

simplify log_{10}(x/(4y))simplify

lo g_{10} (\frac{x}{4 y})

simplify ln(1/(1+7))simplify

ln (\frac{1}{1 + 7})

simplify 1/5 [log_{4}(x)+log_{4}(7)]-[log_{4}(y)]simplify

\frac{1}{5} [lo g_{4} (x) + lo g_{4} (7)] - [lo g_{4} (y)]

Frequently Asked Questions (FAQ)

What is 1/2 ln(9)-3ln(6)+3ln(2) ?
The solution to 1/2 ln(9)-3ln(6)+3ln(2) is -2ln(3)