Solved on Mar 26, 2024

Solve for $x$ in the equation $\ln x^{7} - 4 \ln x = 1$ .

STEP 1

1. The equation $\ln x^{7}-4 \ln x=1$ involves the natural logarithm function, denoted by $\ln$ , which is the inverse of the exponential function with base $e$ .
2. The properties of logarithms, such as the power rule $\ln a^b = b \ln a$ and the quotient rule $\ln a - \ln b = \ln \frac{a}{b}$ , can be used to simplify the equation.
3. The solution will be a positive real number since the logarithm is only defined for positive real numbers.

STEP 2

1. Apply logarithmic properties to combine the logarithmic terms.
2. Isolate the logarithmic term.
3. Exponentiate both sides to solve for $x$ .
4. Simplify the resulting expression to find the exact value of $x$ .

STEP 3

Apply the power rule of logarithms to the term $\ln x^{7}$ .
$\ln x^{7} = 7 \ln x$

STEP 4

Rewrite the equation using the result from STEP_1.
$7 \ln x - 4 \ln x = 1$

STEP 5

Combine the logarithmic terms on the left side of the equation.
$3 \ln x = 1$

STEP 6

Isolate the logarithmic term by dividing both sides of the equation by 3.
$\ln x = \frac{1}{3}$

STEP 7

Exponentiate both sides to remove the logarithm and solve for $x$ .
$e^{\ln x} = e^{\frac{1}{3}}$

STEP 8

Use the property that $e^{\ln a} = a$ to simplify the left side of the equation.
$x = e^{\frac{1}{3}}$

STEP 9

Recognize that $e^{\frac{1}{3}}$ is the cube root of $e$ .
$x = \sqrt[3]{e}$
The solution to the equation is: $x = \sqrt[3]{e}$

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Solve for $x$ in the equation $\ln x^{7} - 4 \ln x = 1$ .

STEP 1

STEP 2

1. Apply logarithmic properties to combine the logarithmic terms.
2. Isolate the logarithmic term.
3. Exponentiate both sides to solve for $x$ .
4. Simplify the resulting expression to find the exact value of $x$ .

STEP 3

Apply the power rule of logarithms to the term $\ln x^{7}$ .
$\ln x^{7} = 7 \ln x$

STEP 4

Rewrite the equation using the result from STEP_1.
$7 \ln x - 4 \ln x = 1$

STEP 5

Combine the logarithmic terms on the left side of the equation.
$3 \ln x = 1$

STEP 6

Isolate the logarithmic term by dividing both sides of the equation by 3.
$\ln x = \frac{1}{3}$

STEP 7

Exponentiate both sides to remove the logarithm and solve for $x$ .
$e^{\ln x} = e^{\frac{1}{3}}$

STEP 8

Use the property that $e^{\ln a} = a$ to simplify the left side of the equation.
$x = e^{\frac{1}{3}}$

STEP 9

Recognize that $e^{\frac{1}{3}}$ is the cube root of $e$ .
$x = \sqrt[3]{e}$
The solution to the equation is: $x = \sqrt[3]{e}$

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Solve for xxx in the equation ln⁡x7−4ln⁡x=1\ln x^{7} - 4 \ln x = 1lnx7−4lnx=1.

STEP 1

STEP 2

1. Apply logarithmic properties to combine the logarithmic terms.2. Isolate the logarithmic term.3. Exponentiate both sides to solve for xxx.4. Simplify the resulting expression to find the exact value of xxx.

STEP 3

Apply the power rule of logarithms to the term ln⁡x7\ln x^{7}lnx7.ln⁡x7=7ln⁡x \ln x^{7} = 7 \ln x lnx7=7lnx

STEP 4

Rewrite the equation using the result from STEP_1.7ln⁡x−4ln⁡x=1 7 \ln x - 4 \ln x = 1 7lnx−4lnx=1

STEP 5

Combine the logarithmic terms on the left side of the equation.3ln⁡x=1 3 \ln x = 1 3lnx=1

STEP 6

Isolate the logarithmic term by dividing both sides of the equation by 3.ln⁡x=13 \ln x = \frac{1}{3} lnx=31​

STEP 7

Exponentiate both sides to remove the logarithm and solve for xxx.eln⁡x=e13 e^{\ln x} = e^{\frac{1}{3}} elnx=e31​

STEP 8

Use the property that eln⁡a=ae^{\ln a} = aelna=a to simplify the left side of the equation.x=e13 x = e^{\frac{1}{3}} x=e31​

STEP 9

Recognize that e13e^{\frac{1}{3}}e31​ is the cube root of eee.x=e3 x = \sqrt[3]{e} x=3e​The solution to the equation is: x=e3 x = \sqrt[3]{e} x=3e​

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Solve for $x$ in the equation $\ln x^{7} - 4 \ln x = 1$ .

1. Apply logarithmic properties to combine the logarithmic terms.
2. Isolate the logarithmic term.
3. Exponentiate both sides to solve for $x$ .
4. Simplify the resulting expression to find the exact value of $x$ .

Apply the power rule of logarithms to the term $\ln x^{7}$ .
$\ln x^{7} = 7 \ln x$

Rewrite the equation using the result from STEP_1.
$7 \ln x - 4 \ln x = 1$

Combine the logarithmic terms on the left side of the equation.
$3 \ln x = 1$

Isolate the logarithmic term by dividing both sides of the equation by 3.
$\ln x = \frac{1}{3}$

Exponentiate both sides to remove the logarithm and solve for $x$ .
$e^{\ln x} = e^{\frac{1}{3}}$

Use the property that $e^{\ln a} = a$ to simplify the left side of the equation.
$x = e^{\frac{1}{3}}$

Recognize that $e^{\frac{1}{3}}$ is the cube root of $e$ .
$x = \sqrt[3]{e}$
The solution to the equation is: $x = \sqrt[3]{e}$