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)

Question

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.

2 \ln 4 x^{3} + 3 \ln y - \frac{1}{3} \ln z^{6}

b.

\ln (x^{2} - 16) - \ln (x + 4)

Answer 1

We know that,
(1)

b \ln a = \ln a^{b}

(2)

\ln a + \ln b = \ln a b

(3)

\ln a - \ln b = \ln \frac{a}{b}

a) Using tha above properties of logarithm, we get

2 \ln 4 x^{3} + 3 \ln y - \frac{1}{3} \ln z^{6}

= \ln (4 x^{3})^{2} + \ln (y)^{3} - \ln (z^{6})^{\frac{1}{3}} [u s i n g p r o p e r t y (1)]

= \ln 16 x^{6} + \ln y^{3} - \ln z^{2}

= 16 x^{6} + \ln \frac{y^{3}}{z^{2}} [u s i n g p r o p e r t y (3)]

= \ln (16 x^{6} \times \frac{y^{3}}{z^{2}}) [u s i n g p r o p e r t y (2)]

= \frac{\ln (16 x^{6} y^{3})}{x^{2}}

∴ 2 \ln 4 x^{3} + 3 \ln y - \frac{1}{3} \ln z^{6} = \frac{\ln (16 x^{6} y^{3})}{z^{2}}

b) Using the above proporties of logarithm, we get

\ln (x 62 - 16) - \ln (x + 4)

= \frac{(x^{2} - 16)}{(x + 4)} [u s i n g p r o p e r t y (3)]

= \frac{\ln ((x + 4) (x - 4))}{x + 4)} [u s i n g (a^{2} - b^{2}) = (a + b) (a - b)]

= \ln (x - 4) [c a n c e l i n g o u t (x + 4) f r o m n u m e r a t o r a n d d e n o m i n a t o r]

∴ \ln (x^{2} - 16) - \ln (x + 4) = \ln (x - 4)

Answer 2

Jeffrey Jordon

Answered 2021-11-03 Author has 2070 answers

Answer is given below (on video)

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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)

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