What is ln((pi/4*47^2)/(pi/4*30^2)) ?

The solution to ln((pi/4*47^2)/(pi/4*30^2)) is ln(pi)-2ln(2)+2ln(47)-ln(pi/4)-2ln(30)

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simplify ln((pi/447^2)/(pi/430^2))

simplify

ln (\frac{\frac{π}{4} \cdot 4 7 ^{2}}{\frac{π}{4} \cdot 3 0 ^{2}})

ln (π) - 2 ln (2) + 2 ln (47) - ln (\frac{π}{4}) - 2 ln (30)

Decimal Notation

0.89790 \dots

Solution steps

ln (\frac{\frac{π}{4} \cdot 4 7 ^{2}}{\frac{π}{4} \cdot 3 0 ^{2}})

Apply log rule:

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

ln (\frac{\frac{π}{4} \cdot 4 7 ^{2}}{\frac{π}{4} \cdot 3 0 ^{2}}) = ln (\frac{π}{4} \cdot 4 7^{2}) - ln (\frac{π}{4} \cdot 3 0^{2})

= ln (4 7^{2} \cdot \frac{π}{4}) - ln (3 0^{2} \cdot \frac{π}{4})

Apply log rule:

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

ln (\frac{π}{4} \cdot 4 7^{2}) = ln (\frac{π}{4}) + ln (4 7^{2}), ln (\frac{π}{4} \cdot 3 0^{2}) = ln (\frac{π}{4}) + ln (3 0^{2})

= ln (\frac{π}{4}) + ln (4 7^{2}) - (ln (3 0^{2}) + ln (\frac{π}{4}))

Apply log rule:

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

ln (4 7^{2}) = 2 ln (47)

= ln (\frac{π}{4}) + 2 ln (47) - (ln (\frac{π}{4}) + ln (3 0^{2}))

Apply log rule:

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

ln (3 0^{2}) = 2 ln (30)

= ln (\frac{π}{4}) + 2 ln (47) - (ln (\frac{π}{4}) + 2 ln (30))

Apply log rule:

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

ln (\frac{π}{4}) = ln (π) - ln (4)

= ln (π) - ln (4) + 2 ln (47) - (2 ln (30) + ln (\frac{π}{4}))

ln (4) = 2 ln (2)

= ln (π) - 2 ln (2) + 2 ln (47) - (2 ln (30) + ln (\frac{π}{4}))

- (ln (\frac{π}{4}) + 2 ln (30)) : - ln (\frac{π}{4}) - 2 ln (30)

= ln (π) - 2 ln (2) + 2 ln (47) - ln (\frac{π}{4}) - 2 ln (30)

lo g_{10} (x^{9} z)

ln (t^{x} (1 - t)^{1 - x})

ln (\frac{x ^{5} ( x - 7 ) ^{6}}{( x ^{2} + 8 ) ^{4}})

- \frac{8.31 \cdot 310}{9.65 \cdot 1 0 ^{4}} \cdot ln (\frac{140}{4})

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

What is ln((pi/4*47^2)/(pi/4*30^2)) ?
The solution to ln((pi/4*47^2)/(pi/4*30^2)) is ln(pi)-2ln(2)+2ln(47)-ln(pi/4)-2ln(30)