What is ln((1.308x10^{-4})/(6.44x10^{-3)}) ?

The solution to ln((1.308x10^{-4})/(6.44x10^{-3)}) is -ln(10)-ln(6.44)+ln(1.308)

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simplify ln((1.308x10^{-4})/(6.44x10^{-3)})

simplify

ln (\frac{1.308 x 1 0 ^{- 4}}{6.44 x 1 0 ^{- 3}})

- ln (10) - ln (6.44) + ln (1.308)

Decimal Notation

- 3.89661 \dots

Solution steps

ln (\frac{1.308 x \cdot 1 0 ^{- 4}}{6.44 x \cdot 1 0 ^{- 3}})

Apply log rule:

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

ln (\frac{1.308 x \cdot 1 0 ^{- 4}}{6.44 x \cdot 1 0 ^{- 3}}) = ln (1.308 x \cdot 1 0^{- 4}) - ln (6.44 x \cdot 1 0^{- 3})

= ln (1 0^{- 4} \cdot 1.308 x) - ln (1 0^{- 3} \cdot 6.44 x)

Apply log rule:

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

ln (1.308 x \cdot 1 0^{- 4}) = ln (1.308) + ln (x) + ln (1 0^{- 4}), ln (6.44 x \cdot 1 0^{- 3}) = ln (6.44) + ln (x) + ln (1 0^{- 3})

= ln (1.308) + ln (x) + ln (1 0^{- 4}) - (ln (x) + ln (6.44) + ln (1 0^{- 3}))

Apply log rule:

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

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

= ln (1.308) + ln (x) - 4 ln (10) - (ln (6.44) + ln (x) + ln (1 0^{- 3}))

Apply log rule:

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

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

= ln (1.308) + ln (x) - 4 ln (10) - (ln (6.44) + ln (x) - 3 ln (10))

- (ln (6.44) + ln (x) - 3 ln (10)) : - ln (6.44) - ln (x) + 3 ln (10)

= ln (1.308) + ln (x) - 4 ln (10) - ln (6.44) - ln (x) + 3 ln (10)

Simplify

ln (1.308) + ln (x) - 4 ln (10) - ln (6.44) - ln (x) + 3 ln (10) : - ln (10) - ln (6.44) + ln (1.308)

= - ln (10) - ln (6.44) + ln (1.308)

ln (x) - ln (a)

2 ln (x) - ln^{2} (x)

5 lo g_{10} (x) - 2 lo g_{10} (x + 1)

lo g_{10} (x) + lo g_{10} (5) - lo g_{10} (3)

What is ln((1.308x10^{-4})/(6.44x10^{-3)}) ?
The solution to ln((1.308x10^{-4})/(6.44x10^{-3)}) is -ln(10)-ln(6.44)+ln(1.308)