What is simplify-5ln(a/c)-1ln(c/a) ?

The solution to simplify-5ln(a/c)-1ln(c/a) is -ln((a^4)/(c^4))

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simplify-5ln(a/c)-1ln(c/a)

simplify

- 5 ln (\frac{a}{c}) - 1 ln (\frac{c}{a})

- ln (\frac{a ^{4}}{c ^{4}})

Solution steps

- 5 ln (\frac{a}{c}) - 1 \cdot 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}) - 1 \cdot ln (\frac{c}{a})

Apply log rule:

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

1 \cdot ln (\frac{c}{a}) = ln ((\frac{c}{a})^{1})

= - ln ((\frac{a}{c})^{5}) - ln ((\frac{c}{a})^{1})

Rewrite

- ln ((\frac{a}{c})^{5}) - ln ((\frac{c}{a})^{1})

- (ln ((\frac{a}{c})^{5}) + ln ((\frac{c}{a})^{1}))

= - (ln ((\frac{a}{c})^{5}) + ln ((\frac{c}{a})^{1}))

Apply log rule:

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

ln ((\frac{a}{c})^{5}) + ln ((\frac{c}{a})^{1}) = ln ((\frac{a}{c})^{5} (\frac{c}{a})^{1})

= - ln ((\frac{a}{c})^{5} (\frac{c}{a})^{1})

Simplify

(\frac{a}{c})^{5} (\frac{c}{a})^{1} : \frac{a ^{4}}{c ^{4}}

= - ln (\frac{a ^{4}}{c ^{4}})

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What is simplify-5ln(a/c)-1ln(c/a) ?
The solution to simplify-5ln(a/c)-1ln(c/a) is -ln((a^4)/(c^4))