What is simplify-5ln(a/c)-8ln(c/a) ?

The solution to simplify-5ln(a/c)-8ln(c/a) is -ln((c^3)/(a^3))

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simplify-5ln(a/c)-8ln(c/a)

simplify

- 5 ln (\frac{a}{c}) - 8 ln (\frac{c}{a})

- ln (\frac{c ^{3}}{a ^{3}})

Solution steps

- 5 ln (\frac{a}{c}) - 8 ln (\frac{c}{a})

Apply log rule:

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

5 ln (\frac{a}{c}) = ln ((\frac{a}{c})^{5})

= - ln ((\frac{a}{c})^{5}) - 8 ln (\frac{c}{a})

Apply log rule:

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

8 ln (\frac{c}{a}) = ln ((\frac{c}{a})^{8})

= - ln ((\frac{a}{c})^{5}) - ln ((\frac{c}{a})^{8})

Rewrite

- ln ((\frac{a}{c})^{5}) - ln ((\frac{c}{a})^{8})

- (ln ((\frac{a}{c})^{5}) + ln ((\frac{c}{a})^{8}))

= - (ln ((\frac{a}{c})^{5}) + ln ((\frac{c}{a})^{8}))

Apply log rule:

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

ln ((\frac{a}{c})^{5}) + ln ((\frac{c}{a})^{8}) = ln ((\frac{c}{a})^{8} (\frac{a}{c})^{5})

= - ln ((\frac{c}{a})^{8} (\frac{a}{c})^{5})

Simplify

(\frac{a}{c})^{5} (\frac{c}{a})^{8} : \frac{c ^{3}}{a ^{3}}

= - ln (\frac{c ^{3}}{a ^{3}})

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ln (x + 6) + ln (x + 1)

What is simplify-5ln(a/c)-8ln(c/a) ?
The solution to simplify-5ln(a/c)-8ln(c/a) is -ln((c^3)/(a^3))