Nash Frank

2022-07-17

a)Express the given quantity as a single logarithm.
$\frac{1}{3}\mathrm{ln}\left(x+2{\right)}^{3}+\frac{1}{2}\left[\mathrm{ln}\left(x\right)-\mathrm{ln}\left({x}^{2}+3x+2{\right)}^{2}\right]$
b) Solve each equation for x.
1)${e}^{5-4x}=4$
x=?
2) $\mathrm{ln}\left(3x-13\right)=8$
x=?

Jeroronryca

Expert

a)Solution:
$\frac{1}{3}\mathrm{ln}\left(x+2{\right)}^{3}+\frac{1}{2}\left[\mathrm{ln}\left(x\right)-\mathrm{ln}\left({x}^{2}+3x+2{\right)}^{2}\right]$
$=\mathrm{ln}\left(x+2{\right)}^{3/3}+\frac{1}{2}\mathrm{ln}x-\frac{1}{2}\mathrm{ln}\left({x}^{2}+3x+2{\right)}^{2}$
$=\mathrm{ln}\left(x+2\right)+\mathrm{ln}{x}^{1/2}-\mathrm{ln}\left({x}^{2}+3x+2{\right)}^{2/2}$
$=\mathrm{ln}\left(x+2\right)\left({x}^{1/2}\right)-\mathrm{ln}\left({x}^{2}+3x+2\right)$
$\left[\because \mathrm{ln}a+\mathrm{ln}b=\mathrm{ln}ab\right]$
$=\mathrm{ln}\frac{\left({x}^{3/2}+2{x}^{1/2}\right)}{\left({x}^{2}+3x+2\right)}$
$\left[\because \mathrm{ln}a-\mathrm{ln}b=\mathrm{ln}\frac{a}{b}\right]$

$=\mathrm{ln}\frac{\left({x}^{3/2}+2{x}^{1/2}\right)}{\left({x}^{2}+3x+2\right)}$

Darryl English

Expert

b)Solution:
1)${e}^{5-4x}=4$
$⇒5-4x=\mathrm{ln}4$
$⇒4x=5-\mathrm{ln}4$
$⇒x=\frac{5}{4}-\frac{1}{4}\mathrm{ln}4$
2)$\mathrm{ln}\left(3x-13\right)=8$
$⇒3x-13={e}^{8}$
$⇒3x=13+{e}^{8}$
math xmlns="http://www.w3.org/1998/Math/MathML"> x = 1 3 ( 13 + e 8 )

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