What is 3log_{10}(a)+5log_{10}(b)-log_{10}(ab) ?

The solution to 3log_{10}(a)+5log_{10}(b)-log_{10}(ab) is log_{10}(a^2b^4)

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simplify 3log_{10}(a)+5log_{10}(b)-log_{10}(ab)

simplify

3 lo g_{10} (a) + 5 lo g_{10} (b) - lo g_{10} (ab)

lo g_{10} (a^{2} b^{4})

Solution steps

3 lo g_{10} (a) + 5 lo g_{10} (b) - lo g_{10} (ab)

3 lo g_{10} (a) = lo g_{10} (a^{3})

5 lo g_{10} (b) = lo g_{10} (b^{5})

= lo g_{10} (a^{3}) + lo g_{10} (b^{5}) - lo g_{10} (ab)

Apply log rule:

lo g_{a} (x) + lo g_{a} (y) = lo g_{a} (x y)

lo g_{10} (a^{3}) + lo g_{10} (b^{5}) = lo g_{10} (a^{3} b^{5})

= lo g_{10} (a^{3} b^{5}) - lo g_{10} (ab)

Apply log rule:

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

lo g_{10} (a^{3} b^{5}) - lo g_{10} (ab) = lo g_{10} (\frac{a ^{3} b ^{5}}{ab})

= lo g_{10} (\frac{a ^{3} b ^{5}}{ab})

Cancel

\frac{a ^{3} b ^{5}}{ab} : a^{2} b^{4}

= lo g_{10} (a^{2} b^{4})

lo g_{10} (5 \cdot 2)

lo g_{5} (\frac{x}{3 ^{2} z})

ln (e^{3 x}) + ln (x^{3}) - ln (3 x)

ln (25 e^{2} \cdot \frac{1}{36})

ln (x^{4}) - ln (x)

What is 3log_{10}(a)+5log_{10}(b)-log_{10}(ab) ?
The solution to 3log_{10}(a)+5log_{10}(b)-log_{10}(ab) is log_{10}(a^2b^4)