What is log_{10}(1000x^4) ?

The solution to log_{10}(1000x^4) is 3+4log_{10}(x)

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simplify log_{10}(1000x^4)

simplify

lo g_{10} (1000 x^{4})

3 + 4 lo g_{10} (x)

Solution steps

lo g_{10} (1000 x^{4})

Factor the number:

1000 = 1 0^{3}

= lo g_{10} (1 0^{3} x^{4})

Apply log rule:

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

lo g_{10} (1 0^{3} x^{4}) = lo g_{10} (1 0^{3}) + lo g_{10} (x^{4})

= lo g_{10} (1 0^{3}) + lo g_{10} (x^{4})

Apply log rule:

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

lo g_{10} (1 0^{3}) = 3

= 3 + lo g_{10} (x^{4})

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x), assuming x > 0

lo g_{10} (x^{4}) = 4 lo g_{10} (x)

= 3 + 4 lo g_{10} (x)

lo g_{10} (\frac{x ^{3}}{y ^{5}})

lo g_{10} (\frac{u ^{4}}{v})

6 lo g_{3} (x) + 2 lo g_{3} (11)

lo g_{4} (\frac{\frac{x}{z}}{w})

lo g_{10} (20) + lo g_{10} (15) - 2 lo g_{10} (5)