Solved on Jan 21, 2024

Solve the equation 7ln(x)ln(x2)=17 \ln (x) - \ln (x^{2}) = 1 for the unknown variable xx.

STEP 1

Assumptions
1. We are given the equation 7ln(x)ln(x2)=17 \ln (x) - \ln \left(x^{2}\right) = 1.
2. We assume that x>0x > 0 since the natural logarithm is only defined for positive numbers.

STEP 2

Use the properties of logarithms to combine the logarithmic terms on the left side of the equation. The property ln(a)ln(b)=ln(ab)\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) can be applied here.
7ln(x)ln(x2)=ln(x7)ln(x2)7 \ln (x) - \ln \left(x^{2}\right) = \ln\left(x^{7}\right) - \ln\left(x^{2}\right)

STEP 3

Apply the property mentioned in STEP_2 to combine the logarithms into a single logarithm.
ln(x7)ln(x2)=ln(x7x2)\ln\left(x^{7}\right) - \ln\left(x^{2}\right) = \ln\left(\frac{x^{7}}{x^{2}}\right)

STEP 4

Simplify the expression inside the logarithm using the exponent rule xaxb=xab\frac{x^{a}}{x^{b}} = x^{a-b}.
ln(x7x2)=ln(x72)=ln(x5)\ln\left(\frac{x^{7}}{x^{2}}\right) = \ln\left(x^{7-2}\right) = \ln\left(x^{5}\right)

STEP 5

Now we have the simplified equation:
ln(x5)=1\ln\left(x^{5}\right) = 1

STEP 6

To solve for xx, we need to get rid of the logarithm. We can exponentiate both sides of the equation with base ee to do this because ln(x)=y\ln(x) = y is equivalent to ey=xe^{y} = x.
eln(x5)=e1e^{\ln\left(x^{5}\right)} = e^{1}

STEP 7

Simplify the left side using the property that eln(a)=ae^{\ln(a)} = a.
x5=e1x^{5} = e^{1}

STEP 8

Since e1=ee^{1} = e, we have:
x5=ex^{5} = e

STEP 9

To solve for xx, take the fifth root of both sides of the equation.
x=e5x = \sqrt[5]{e}

STEP 10

The fifth root of ee is the solution for xx.
x=e5x = \sqrt[5]{e}
Therefore, the solution to the equation 7ln(x)ln(x2)=17 \ln (x) - \ln \left(x^{2}\right) = 1 is x=e5x = \sqrt[5]{e}.

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