Solved on Mar 26, 2024

Solve for xx in the equation lnx74lnx=1\ln x^{7} - 4 \ln x = 1.

STEP 1

1. The equation lnx74lnx=1\ln x^{7}-4 \ln x=1 involves the natural logarithm function, denoted by ln\ln, which is the inverse of the exponential function with base ee.
2. The properties of logarithms, such as the power rule lnab=blna\ln a^b = b \ln a and the quotient rule lnalnb=lnab\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 xx.
4. Simplify the resulting expression to find the exact value of xx.

STEP 3

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

STEP 4

Rewrite the equation using the result from STEP_1.
7lnx4lnx=1 7 \ln x - 4 \ln x = 1

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

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