What is ln(10000)-18ln(500) ?

The solution to ln(10000)-18ln(500) is -2ln(5^{25}*65536)

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simplify ln(10000)-18ln(500)

simplify

ln (10000) - 18 ln (500)

- 2 ln (5^{25} \cdot 65536)

Decimal Notation

- 102.65260 \dots

Solution steps

ln (10000) - 18 ln (500)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

18 ln (500) = ln (50 0^{18})

= ln (10000) - ln (50 0^{18})

Apply log rule:

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

ln (10000) - ln (50 0^{18}) = ln (\frac{10000}{50 0 ^{18}})

= ln (\frac{10000}{50 0 ^{18}})

\frac{10000}{50 0 ^{18}} = \frac{1}{5 ^{50} \cdot 2 ^{32}}

= ln (\frac{1}{5 ^{50} \cdot 2 ^{32}})

Apply log rule:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

= - ln (5^{50} \cdot 2^{32})

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

5^{50} \cdot 2^{32} = (5^{25} \cdot 2^{16})^{2}

= - ln ((5^{25} \cdot 2^{16})^{2})

Apply log rule

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

assuming

x \geq 0

= - 2 ln (5^{25} \cdot 2^{16})

2^{16} = 65536

= - 2 ln (5^{25} \cdot 65536)

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ln (\frac{x ^{2}}{e ^{3}})

What is ln(10000)-18ln(500) ?
The solution to ln(10000)-18ln(500) is -2ln(5^{25}*65536)