What is ln((1.5*10^4)/(1.5*10^{10)}) ?

The solution to ln((1.5*10^4)/(1.5*10^{10)}) is -6ln(10)

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simplify ln((1.510^4)/(1.510^{10)})

simplify

ln (\frac{1.5 \cdot 1 0 ^{4}}{1.5 \cdot 1 0 ^{10}})

- 6 ln (10)

Decimal Notation

- 13.81551 \dots

Solution steps

ln (\frac{1.5 \cdot 1 0 ^{4}}{1.5 \cdot 1 0 ^{10}})

Apply log rule:

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

ln (\frac{1.5 \cdot 1 0 ^{4}}{1.5 \cdot 1 0 ^{10}}) = ln (1.5 \cdot 1 0^{4}) - ln (1.5 \cdot 1 0^{10})

= ln (1 0^{4} \cdot 1.5) - ln (1 0^{10} \cdot 1.5)

Apply log rule:

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

ln (1.5 \cdot 1 0^{4}) = ln (1.5) + ln (1 0^{4}), ln (1.5 \cdot 1 0^{10}) = ln (1.5) + ln (1 0^{10})

= ln (1.5) + ln (1 0^{4}) - (ln (1.5) + ln (1 0^{10}))

Apply log rule:

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

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

= ln (1.5) + 4 ln (10) - (ln (1.5) + ln (1 0^{10}))

Apply log rule:

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

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

= ln (1.5) + 4 ln (10) - (ln (1.5) + 10 ln (10))

- (ln (1.5) + 10 ln (10)) : - ln (1.5) - 10 ln (10)

= ln (1.5) + 4 ln (10) - ln (1.5) - 10 ln (10)

Simplify

ln (1.5) + 4 ln (10) - ln (1.5) - 10 ln (10) : - 6 ln (10)

= - 6 ln (10)

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What is ln((1.5*10^4)/(1.5*10^{10)}) ?
The solution to ln((1.5*10^4)/(1.5*10^{10)}) is -6ln(10)