What is ln(5)-2(ln(4))+ln(32) ?

The solution to ln(5)-2(ln(4))+ln(32) is ln(10)

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simplify ln(5)-2(ln(4))+ln(32)

simplify

ln (5) - 2 (ln (4)) + ln (32)

ln (10)

Decimal Notation

2.30258 \dots

Solution steps

ln (5) - 2 (ln (4)) + ln (32)

Apply log rule:

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

ln (5) + ln (32) = ln (5 \cdot 32)

= ln (5 \cdot 32) - 2 ln (4)

Multiply the numbers:

5 \cdot 32 = 160

= ln (160) - 2 ln (4)

Apply log rule:

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

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

= ln (160) - ln (4^{2})

Apply log rule:

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

ln (160) - ln (4^{2}) = ln (\frac{160}{4 ^{2}})

= ln (\frac{160}{4 ^{2}})

\frac{160}{4 ^{2}} = 10

= ln (10)

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