What is ln|7-2|-3/2 ln|2*7-1|+2ln|7+3| ?

The solution to ln|7-2|-3/2 ln|2*7-1|+2ln|7+3| is ln((500sqrt(13))/(169))

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simplify ln|7-2|-3/2 ln|2*7-1|+2ln|7+3|

simplify

ln ∣ 7 - 2 ∣ - \frac{3}{2} ln ∣ 2 \cdot 7 - 1 ∣ + 2 ln ∣ 7 + 3 ∣

ln (\frac{500 13}{169})

Decimal Notation

2.36718 \dots

Solution steps

ln ∣ 7 - 2 ∣ - \frac{3}{2} ln ∣ 2 \cdot 7 - 1 ∣ + 2 ln ∣ 7 + 3 ∣

Apply log rule:

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

\frac{3}{2} ln ∣ 2 \cdot 7 - 1 ∣ = ln (∣ 2 \cdot 7 - 1 ∣^{\frac{3}{2}})

= ln ∣ 7 - 2 ∣ - ln (∣ 2 \cdot 7 - 1 ∣^{\frac{3}{2}}) + 2 ln ∣ 7 + 3 ∣

Apply log rule:

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

ln ∣ 7 - 2 ∣ - ln (∣ 2 \cdot 7 - 1 ∣^{\frac{3}{2}}) = ln (\frac{∣ 7 - 2 ∣}{∣ 2 \cdot 7 - 1 ∣ ^{\frac{3}{2}}})

= ln (\frac{∣ 7 - 2 ∣}{∣ 2 \cdot 7 - 1 ∣ ^{\frac{3}{2}}}) + 2 ln ∣ 7 + 3 ∣

\frac{∣ 7 - 2 ∣}{∣ 2 \cdot 7 - 1 ∣ ^{\frac{3}{2}}} = \frac{5 13}{169}

= ln (\frac{5 13}{169}) + 2 ln ∣ 7 + 3 ∣

Apply log rule:

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

2 ln ∣ 7 + 3 ∣ = ln (∣ 7 + 3 ∣^{2})

= ln (\frac{5 13}{169}) + ln (∣ 7 + 3 ∣^{2})

Apply log rule:

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

ln (\frac{5 13}{169}) + ln (∣ 7 + 3 ∣^{2}) = ln (\frac{5 13}{169} ∣ 7 + 3 ∣^{2})

= ln (\frac{5 13}{169} ∣ 7 + 3 ∣^{2})

\frac{5 13}{169} ∣ 7 + 3 ∣^{2} = \frac{500 13}{169}

= ln (\frac{500 13}{169})

lo g_{10} (\frac{2 x ^{2}}{( y - 1 ) ^{3}})

\frac{1}{50} ln (\frac{1260}{980})

ln (8 e^{8 x})

2 (lo g_{10} (x + 1) - lo g_{10} (x - 4))

- (\frac{3}{11}) lo g_{2} (\frac{3}{11}) - (\frac{8}{11}) lo g_{2} (\frac{8}{11})

What is ln|7-2|-3/2 ln|2*7-1|+2ln|7+3| ?
The solution to ln|7-2|-3/2 ln|2*7-1|+2ln|7+3| is ln((500sqrt(13))/(169))