What is 3log_{10}(ts)-4log_{10}(s/t) ?

The solution to 3log_{10}(ts)-4log_{10}(s/t) is log_{10}((t^7)/s)

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simplify 3log_{10}(ts)-4log_{10}(s/t)

simplify

3 lo g_{10} (t s) - 4 lo g_{10} (\frac{s}{t})

lo g_{10} (\frac{t ^{7}}{s})

Solution steps

3 lo g_{10} (t s) - 4 lo g_{10} (\frac{s}{t})

3 lo g_{10} (t s) = lo g_{10} ((t s)^{3})

- 4 lo g_{10} (\frac{s}{t}) = - lo g_{10} ((\frac{s}{t})^{4})

= lo g_{10} ((t s)^{3}) - lo g_{10} ((\frac{s}{t})^{4})

Apply log rule:

lo g_{a} (x) - lo g_{a} (y) = lo g_{a} (\frac{x}{y})

lo g_{10} ((t s)^{3}) - lo g_{10} ((\frac{s}{t})^{4}) = lo g_{10} (\frac{( t s ) ^{3}}{( \frac{s}{t} ) ^{4}})

= lo g_{10} (\frac{( t s ) ^{3}}{( \frac{s}{t} ) ^{4}})

Simplify

\frac{( t s ) ^{3}}{( \frac{s}{t} ) ^{4}} : \frac{t ^{7}}{s}

= lo g_{10} (\frac{t ^{7}}{s})

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What is 3log_{10}(ts)-4log_{10}(s/t) ?
The solution to 3log_{10}(ts)-4log_{10}(s/t) is log_{10}((t^7)/s)