Use the properties of logarithms to rewrite each expression as the logarithm of a single expression. Be sure to use positive exponents and avoid radicals. a. 2ln4x^{3} + 3ln y - frac{1}{3}ln z^{6} b. ln(x^{2} - 16) - ln(x + 4)

Brittney Lord 2020-11-09 Answered
Use the properties of logarithms to rewrite each expression as the logarithm of a single expression. Be sure to use positive exponents and avoid radicals.
a. 2ln4x3 + 3ln y  13ln z6
b. ln(x2  16)  ln(x + 4)
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Expert Answer

Latisha Oneil
Answered 2020-11-10 Author has 100 answers
We know that,
(1) bln a=lnab
(2) ln a + ln b=ln ab
(3) ln a  ln b=ln ab
a) Using tha above properties of logarithm, we get
2ln 4x3 + 3ln y  13lnz6
=ln(4x3)2 + ln(y)3  ln(z6)13 [using property (1)]
=ln 16x6 + ln y3  lnz2
=16x6 + ln y3z2 [using property (3)]
=ln (16x6 × y3z2) [using property (2)]
=ln(16x6y3)x2
 2 ln 4x3 + 3 ln y  13ln z6=ln(16x6y3)z2
b) Using the above proporties of logarithm, we get
ln(x62  16)  ln(x + 4)
=(x2  16)(x + 4) [using property (3)]
=ln((x + 4)(x  4))x + 4) [using (a2  b2)=(a + b)(a  b)]
=ln(x  4) [canceling out (x + 4) from numerator and denominator]
 ln(x2  16)  ln(x + 4)=ln(x  4)
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Jeffrey Jordon
Answered 2021-11-03 Author has 2070 answers

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