a)Express the given quantity as a single logarithm. 1 3 ln ⁡ ( x +...

Nash Frank

Nash Frank

Answered

2022-07-17

a)Express the given quantity as a single logarithm.
1 3 ln ( x + 2 ) 3 + 1 2 [ ln ( x ) ln ( x 2 + 3 x + 2 ) 2 ]
b) Solve each equation for x.
1) e 5 4 x = 4
x=?
2) ln ( 3 x 13 ) = 8
x=?

Answer & Explanation

Jeroronryca

Jeroronryca

Expert

2022-07-18Added 13 answers

a)Solution:
1 3 ln ( x + 2 ) 3 + 1 2 [ ln ( x ) ln ( x 2 + 3 x + 2 ) 2 ]
= ln ( x + 2 ) 3 / 3 + 1 2 ln x 1 2 ln ( x 2 + 3 x + 2 ) 2
= ln ( x + 2 ) + ln x 1 / 2 ln ( x 2 + 3 x + 2 ) 2 / 2
= ln ( x + 2 ) ( x 1 / 2 ) ln ( x 2 + 3 x + 2 )
[ ln a + ln b = ln a b ]
= ln ( x 3 / 2 + 2 x 1 / 2 ) ( x 2 + 3 x + 2 )
[ ln a ln b = ln a b ]
Hence    1 3 ln ( x + 2 ) 3 + 1 2 [ ln x ln ( x 2 + 3 x + 2 ) 2 ]
= ln ( x 3 / 2 + 2 x 1 / 2 ) ( x 2 + 3 x + 2 )
Darryl English

Darryl English

Expert

2022-07-19Added 2 answers

b)Solution:
1) e 5 4 x = 4
5 4 x = ln 4
4 x = 5 ln 4
x = 5 4 1 4 ln 4
2) ln ( 3 x 13 ) = 8
3 x 13 = e 8
3 x = 13 + e 8
math xmlns="http://www.w3.org/1998/Math/MathML"> x = 1 3 ( 13 + e 8 )

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