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Solved on Dec 09, 2023

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

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

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