What is (log_{e}(a^b)-log_{e}(a^c))/(b-c) ?

The solution to (log_{e}(a^b)-log_{e}(a^c))/(b-c) is (ln(a^{b-c}))/(b-c)

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simplify (log_{e}(a^b)-log_{e}(a^c))/(b-c)

simplify

\frac{lo g _{e} ( a ^{b} ) - lo g _{e} ( a ^{c} )}{b - c}

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

Solution steps

\frac{lo g _{e} ( a ^{b} ) - lo g _{e} ( a ^{c} )}{b - c}

Apply log rule:

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

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

= \frac{lo g _{e} ( \frac{a ^{b}}{a ^{c}} )}{b - c}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{a ^{b}}{a ^{c}} = a^{b - c}

= \frac{lo g _{e} ( a ^{b - c} )}{b - c}

Simplify

lo g_{e} (a^{b - c}) : ln (a^{b - c})

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

\frac{lo g _{10} ( 9 x )}{2} - lo g_{10} (\frac{2}{3 x})

ln (\frac{x}{c})

ln (\frac{3}{20})

lo g_{5} (\frac{5 ^{x} x ^{5}}{5 ^{z} z ^{5}})

ln (\frac{13}{12})

What is (log_{e}(a^b)-log_{e}(a^c))/(b-c) ?
The solution to (log_{e}(a^b)-log_{e}(a^c))/(b-c) is (ln(a^{b-c}))/(b-c)