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

Studdy Solution

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

Studdy Solution

STEP 1

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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Question Simplify ln⁡xe2x\ln \frac{x}{e^{2 x}}lne2xx​ by expressing it as a sum or difference of logarithms, and represent powers as factors.

Studdy Solution

STEP 1

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)ln(e2xx​)=ln(x)−ln(e2x)

STEP 3

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

STEP 4

Since the natural logarithm of eee is 1, ln⁡(e)=1\ln(e) = 1ln(e)=1, we can simplify the second term.2x⋅ln⁡(e)=2x⋅1=2x2 x \cdot \ln(e) = 2 x \cdot 1 = 2 x2x⋅ln(e)=2x⋅1=2x

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 xln(x)−ln(e2x)=ln(x)−2x

STEP 6

The expression ln⁡xe2x\ln \frac{x}{e^{2 x}}lne2xx​ as a sum and/or difference of logarithms, with powers expressed as factors, is:ln⁡(x)−2x\ln(x) - 2 xln(x)−2x

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

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

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

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

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

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