What is ln(x)+6ln((y+x)/x)-9ln((y+2x)/x) ?

The solution to ln(x)+6ln((y+x)/x)-9ln((y+2x)/x) is ln((x^4(y+x)^6)/((y+2x)^9))

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

simplify ln(x)+6ln((y+x)/x)-9ln((y+2x)/x)

simplify

ln (x) + 6 ln (\frac{y + x}{x}) - 9 ln (\frac{y + 2 x}{x})

ln (\frac{x ^{4} ( y + x ) ^{6}}{( y + 2 x ) ^{9}})

Solution steps

ln (x) + 6 ln (\frac{y + x}{x}) - 9 ln (\frac{y + 2 x}{x})

Apply log rule:

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

6 ln (\frac{x + y}{x}) = ln ((\frac{x + y}{x})^{6})

= ln (x) + ln ((\frac{x + y}{x})^{6}) - 9 ln (\frac{2 x + y}{x})

Apply log rule:

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

ln (x) + ln ((\frac{y + x}{x})^{6}) = ln ((\frac{x + y}{x})^{6} x)

= ln ((\frac{x + y}{x})^{6} x) - 9 ln (\frac{2 x + y}{x})

Simplify

x (\frac{y + x}{x})^{6} : \frac{( y + x ) ^{6}}{x ^{5}}

= ln (\frac{( y + x ) ^{6}}{x ^{5}}) - 9 ln (\frac{y + 2 x}{x})

Apply log rule:

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

9 ln (\frac{2 x + y}{x}) = ln ((\frac{2 x + y}{x})^{9})

= ln (\frac{( x + y ) ^{6}}{x ^{5}}) - ln ((\frac{2 x + y}{x})^{9})

Apply log rule:

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

ln (\frac{( y + x ) ^{6}}{x ^{5}}) - ln ((\frac{y + 2 x}{x})^{9}) = ln (\frac{\frac{( x + y ) ^{6}}{x ^{5}}}{( \frac{2 x + y}{x} ) ^{9}})

= ln (\frac{\frac{( y + x ) ^{6}}{x ^{5}}}{( \frac{y + 2 x}{x} ) ^{9}})

Simplify

\frac{\frac{( y + x ) ^{6}}{x ^{5}}}{( \frac{y + 2 x}{x} ) ^{9}} : \frac{x ^{4} ( y + x ) ^{6}}{( y + 2 x ) ^{9}}

= ln (\frac{x ^{4} ( y + x ) ^{6}}{( y + 2 x ) ^{9}})

- 7 ln (7 \cdot 0.9)

ln (112) - ln (133)

lo g_{10} (1.04167) + 0.6 \cdot lo g_{10} (0.749996)

lo g_{2} ((x^{3} + 2) (x - 5)^{3})

\frac{1}{0.0105} ln (\frac{6000}{5000})

What is ln(x)+6ln((y+x)/x)-9ln((y+2x)/x) ?
The solution to ln(x)+6ln((y+x)/x)-9ln((y+2x)/x) is ln((x^4(y+x)^6)/((y+2x)^9))