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Expand the logarithmic expression $\ln\left(\frac{\left(3x+22\right)^2\left(-9+e^{2x}\right)}{\sqrt[4]{\left(6x^2+2\right)^3}}\right)$

Step-by-step Solution

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Final Answer

$2\ln\left(3x+22\right)+\ln\left(-9+e^{2x}\right)-\frac{3}{4}\ln\left(6x^2+2\right)$
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Step-by-step Solution

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The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator

$\ln\left(\left(3x+22\right)^2\left(-9+e^{2x}\right)\right)-\ln\left(\sqrt[4]{\left(6x^2+2\right)^3}\right)$

Learn how to solve expanding logarithms problems step by step online.

$\ln\left(\left(3x+22\right)^2\left(-9+e^{2x}\right)\right)-\ln\left(\sqrt[4]{\left(6x^2+2\right)^3}\right)$

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Learn how to solve expanding logarithms problems step by step online. Expand the logarithmic expression ln(((3x+22)^2(-9+e^(2x)))/((6x^2+2)^3^1/4)). The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator. Using the power rule of logarithms: \log_a(x^n)=n\cdot\log_a(x). Using the power rule of logarithms: \log_a(x^n)=n\cdot\log_a(x). Applying the product rule for logarithms: \log_b\left(MN\right)=\log_b\left(M\right)+\log_b\left(N\right).

Final Answer

$2\ln\left(3x+22\right)+\ln\left(-9+e^{2x}\right)-\frac{3}{4}\ln\left(6x^2+2\right)$

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SimplifyWrite as single logarithmCondense the logarithmExpand the logarithmFind the integralFind the derivativeSolve for x

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Function Plot

Plotting: $2\ln\left(3x+22\right)+\ln\left(-9+e^{2x}\right)-\frac{3}{4}\ln\left(6x^2+2\right)$

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7
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9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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Main Topic: Expanding Logarithms

Logarithm expansion consists of applying the properties of logarithms to express a single logarithm in multiple logarithms, usually much simpler.

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