What is 5ln(y/x+1)-4ln(y/x+2)+ln(x) ?

The solution to 5ln(y/x+1)-4ln(y/x+2)+ln(x) is ln(((y+x)^5)/((y+2x)^4))

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simplify 5ln(y/x+1)-4ln(y/x+2)+ln(x)

simplify

5 ln (\frac{y}{x} + 1) - 4 ln (\frac{y}{x} + 2) + ln (x)

ln (\frac{( y + x ) ^{5}}{( y + 2 x ) ^{4}})

Solution steps

5 ln (\frac{y}{x} + 1) - 4 ln (\frac{y}{x} + 2) + ln (x)

Apply log rule:

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

5 ln (\frac{y}{x} + 1) = ln ((\frac{y}{x} + 1)^{5})

= ln ((\frac{y}{x} + 1)^{5}) - 4 ln (\frac{y}{x} + 2) + ln (x)

Apply log rule:

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

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

= ln ((\frac{y}{x} + 1)^{5}) - ln ((\frac{y}{x} + 2)^{4}) + ln (x)

Apply log rule:

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

ln ((\frac{y}{x} + 1)^{5}) - ln ((\frac{y}{x} + 2)^{4}) = ln (\frac{( \frac{y}{x} + 1 ) ^{5}}{( \frac{y}{x} + 2 ) ^{4}})

= ln (\frac{( \frac{y}{x} + 1 ) ^{5}}{( \frac{y}{x} + 2 ) ^{4}}) + ln (x)

Simplify

\frac{( \frac{y}{x} + 1 ) ^{5}}{( \frac{y}{x} + 2 ) ^{4}} : \frac{( y + x ) ^{5}}{x ( y + 2 x ) ^{4}}

= ln (\frac{( y + x ) ^{5}}{x ( y + 2 x ) ^{4}}) + ln (x)

Apply log rule:

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

ln (\frac{( y + x ) ^{5}}{x ( y + 2 x ) ^{4}}) + ln (x) = ln (x \frac{( y + x ) ^{5}}{x ( y + 2 x ) ^{4}})

= ln (x \frac{( y + x ) ^{5}}{x ( y + 2 x ) ^{4}})

Simplify

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

= ln (\frac{( y + x ) ^{5}}{( y + 2 x ) ^{4}})

ln (9) - 3 ln (x) - 2 ln (y)

lo g_{10} (2) + 3 lo g_{10} (x) - \frac{1}{2} lo g_{10} (y)

ln (\frac{4.1 \cdot 1 0 ^{8}}{1.7 \cdot 1 0 ^{7}})

3 ln (\frac{11}{10.5})

2 lo g_{c} (m) - lo g_{c} (n)

What is 5ln(y/x+1)-4ln(y/x+2)+ln(x) ?
The solution to 5ln(y/x+1)-4ln(y/x+2)+ln(x) is ln(((y+x)^5)/((y+2x)^4))