What is log_{10}((5*10^{-6})/(1*10^{-12)}) ?

The solution to log_{10}((5*10^{-6})/(1*10^{-12)}) is log_{10}(5)+6

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simplify log_{10}((510^{-6})/(110^{-12)})

simplify

lo g_{10} (\frac{5 \cdot 1 0 ^{- 6}}{1 \cdot 1 0 ^{- 12}})

lo g_{10} (5) + 6

Decimal Notation

6.69897 \dots

Solution steps

lo g_{10} (\frac{5 \cdot 1 0 ^{- 6}}{1 \cdot 1 0 ^{- 12}})

Apply log rule:

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

lo g_{10} (\frac{5 \cdot 1 0 ^{- 6}}{1 \cdot 1 0 ^{- 12}}) = lo g_{10} (5 \cdot 1 0^{- 6}) - lo g_{10} (1 \cdot 1 0^{- 12})

= lo g_{10} (1 0^{- 6} \cdot 5) - lo g_{10} (1 0^{- 12} \cdot 1)

Apply log rule:

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

lo g_{10} (5 \cdot 1 0^{- 6}) = lo g_{10} (5) + lo g_{10} (1 0^{- 6}), lo g_{10} (1 \cdot 1 0^{- 12}) = lo g_{10} (1) + lo g_{10} (1 0^{- 12})

= lo g_{10} (5) + lo g_{10} (1 0^{- 6}) - (lo g_{10} (1) + lo g_{10} (1 0^{- 12}))

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

lo g_{10} (1 0^{- 6}) = - 6 lo g_{10} (10)

= lo g_{10} (5) - 6 lo g_{10} (10) - (lo g_{10} (1) + lo g_{10} (1 0^{- 12}))

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

lo g_{10} (1 0^{- 12}) = - 12 lo g_{10} (10)

= lo g_{10} (5) - 6 lo g_{10} (10) - (lo g_{10} (1) - 12 lo g_{10} (10))

6 lo g_{10} (10) = 6

- (lo g_{10} (1) - 12 lo g_{10} (10)) = 12

= lo g_{10} (5) - 6 + 12

Add/Subtract the numbers:

- 6 + 12 = 6

= lo g_{10} (5) + 6

(\frac{1}{2})^{2} - \frac{1}{2} - ln (\frac{1}{2})

ln (\frac{16384 x ^{12} + 4}{0.1})

ln (0.023) - ln (0.02)

- lo g_{10} (\frac{1 0 ^{- 14}}{\frac{1}{30}})

- lo g_{10} (\frac{5}{100})

What is log_{10}((5*10^{-6})/(1*10^{-12)}) ?
The solution to log_{10}((5*10^{-6})/(1*10^{-12)}) is log_{10}(5)+6