What is ln((5*10^{-3})/(5*10^{-4)}) ?

The solution to ln((5*10^{-3})/(5*10^{-4)}) is ln(10)

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simplify ln((510^{-3})/(510^{-4)})

simplify

ln (\frac{5 \cdot 1 0 ^{- 3}}{5 \cdot 1 0 ^{- 4}})

ln (10)

Decimal Notation

2.30258 \dots

Solution steps

ln (\frac{5 \cdot 1 0 ^{- 3}}{5 \cdot 1 0 ^{- 4}})

Apply log rule:

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

ln (\frac{5 \cdot 1 0 ^{- 3}}{5 \cdot 1 0 ^{- 4}}) = ln (5 \cdot 1 0^{- 3}) - ln (5 \cdot 1 0^{- 4})

= ln (1 0^{- 3} \cdot 5) - ln (1 0^{- 4} \cdot 5)

Apply log rule:

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

ln (5 \cdot 1 0^{- 3}) = ln (5) + ln (1 0^{- 3}), ln (5 \cdot 1 0^{- 4}) = ln (5) + ln (1 0^{- 4})

= ln (5) + ln (1 0^{- 3}) - (ln (5) + ln (1 0^{- 4}))

Apply log rule:

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

ln (1 0^{- 3}) = - 3 ln (10)

= ln (5) - 3 ln (10) - (ln (5) + ln (1 0^{- 4}))

Apply log rule:

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

ln (1 0^{- 4}) = - 4 ln (10)

= ln (5) - 3 ln (10) - (ln (5) - 4 ln (10))

- (ln (5) - 4 ln (10)) : - ln (5) + 4 ln (10)

= ln (5) - 3 ln (10) - ln (5) + 4 ln (10)

Simplify

ln (5) - 3 ln (10) - ln (5) + 4 ln (10) : ln (10)

= ln (10)

\frac{1}{12} (ln \frac{3}{2} x + 1 - ln \frac{3}{2} x - 1) + C

ln n^{2} + 5 - ln ∣ 14 ∣

ln (\frac{7}{17})

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

lo g_{10} (\frac{5 k}{n ^{3}})

What is ln((5*10^{-3})/(5*10^{-4)}) ?
The solution to ln((5*10^{-3})/(5*10^{-4)}) is ln(10)