Solved on Dec 09, 2023

Simplify lnxe2x\ln \frac{x}{e^{2 x}} by expressing it as a sum or difference of logarithms, and represent powers as factors.

STEP 1

Assumptions
1. The expression to simplify is lnxe2x\ln \frac{x}{e^{2 x}}.
2. The properties of logarithms that we will use are: - The logarithm of a quotient is the difference of the logarithms: ln(ab)=ln(a)ln(b)\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b). - The logarithm of a product is the sum of the logarithms: ln(ab)=ln(a)+ln(b)\ln(ab) = \ln(a) + \ln(b). - The power rule for logarithms: ln(ab)=bln(a)\ln(a^b) = b \cdot \ln(a).

STEP 2

Apply the logarithm quotient rule to the given expression.
ln(xe2x)=ln(x)ln(e2x)\ln \left(\frac{x}{e^{2 x}}\right) = \ln(x) - \ln\left(e^{2 x}\right)

STEP 3

Apply the power rule for logarithms to the second term.
ln(e2x)=2xln(e)\ln\left(e^{2 x}\right) = 2 x \cdot \ln(e)

STEP 4

Since the natural logarithm of ee is 1, ln(e)=1\ln(e) = 1, we can simplify the second term.
2xln(e)=2x1=2x2 x \cdot \ln(e) = 2 x \cdot 1 = 2 x

STEP 5

Substitute the simplified second term back into the expression.
ln(x)ln(e2x)=ln(x)2x\ln(x) - \ln\left(e^{2 x}\right) = \ln(x) - 2 x

STEP 6

The expression lnxe2x\ln \frac{x}{e^{2 x}} as a sum and/or difference of logarithms, with powers expressed as factors, is:
ln(x)2x\ln(x) - 2 x

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