What is ln((1.5*10^5)/(1.5*10^{10)}) ?

The solution to ln((1.5*10^5)/(1.5*10^{10)}) is -5ln(10)

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simplify ln((1.510^5)/(1.510^{10)})

simplify

ln (\frac{1.5 \cdot 1 0 ^{5}}{1.5 \cdot 1 0 ^{10}})

- 5 ln (10)

Decimal Notation

- 11.51292 \dots

Solution steps

ln (\frac{1.5 \cdot 1 0 ^{5}}{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 ^{5}}{1.5 \cdot 1 0 ^{10}}) = ln (1.5 \cdot 1 0^{5}) - ln (1.5 \cdot 1 0^{10})

= ln (1 0^{5} \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^{5}) = ln (1.5) + ln (1 0^{5}), ln (1.5 \cdot 1 0^{10}) = ln (1.5) + ln (1 0^{10})

= ln (1.5) + ln (1 0^{5}) - (ln (1.5) + ln (1 0^{10}))

Apply log rule:

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

ln (1 0^{5}) = 5 ln (10)

= ln (1.5) + 5 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) + 5 ln (10) - (ln (1.5) + 10 ln (10))

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

= ln (1.5) + 5 ln (10) - ln (1.5) - 10 ln (10)

Simplify

ln (1.5) + 5 ln (10) - ln (1.5) - 10 ln (10) : - 5 ln (10)

= - 5 ln (10)

lo g_{10} (1024) - 3 lo g_{10} (2) - lo g_{10} (3)

lo g_{10} (t^{3})

ln (3 x) + ln (2)

ln (\frac{n + 1}{n - 1}) - ln (n)

lo g_{2} (\frac{x}{y ^{4}})

What is ln((1.5*10^5)/(1.5*10^{10)}) ?
The solution to ln((1.5*10^5)/(1.5*10^{10)}) is -5ln(10)