Solved on Feb 13, 2024

Solve the equation $8|\ln x|-16=0$ . The solution set is $\{e^{2}, e^{-2}\}$ .

STEP 1

Assumptions
1. We are given the equation $8|\ln x| - 16 = 0$ .
2. We need to solve for $x$ in terms of the natural number $e$ .
3. The absolute value function $|y|$ splits the equation into two cases, $y$ and $-y$ .
4. The natural logarithm function $\ln x$ is defined for $x > 0$ .

STEP 2

Isolate the absolute value expression by adding $16$ to both sides of the equation.
$8|\ln x| - 16 + 16 = 0 + 16$

STEP 3

Simplify the equation.
$8|\ln x| = 16$

STEP 4

Divide both sides of the equation by $8$ to solve for $|\ln x|$ .
$\frac{8|\ln x|}{8} = \frac{16}{8}$

STEP 5

Simplify the equation further.
$|\ln x| = 2$

STEP 6

Now we need to consider the two cases for the absolute value.
Case 1: $\ln x = 2$
Case 2: $\ln x = -2$

STEP 7

Solve for $x$ in Case 1 by exponentiating both sides of the equation to eliminate the natural logarithm.
$e^{\ln x} = e^2$

STEP 8

Simplify using the property $e^{\ln a} = a$ .
$x = e^2$

STEP 9

Solve for $x$ in Case 2 by exponentiating both sides of the equation to eliminate the natural logarithm.
$e^{\ln x} = e^{-2}$

STEP 10

Simplify using the property $e^{\ln a} = a$ .
$x = e^{-2}$

STEP 11

Combine the solutions from Case 1 and Case 2.
The solution set is $\{e^2, e^{-2}\}$ .

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Solve the equation $8|\ln x|-16=0$ . The solution set is $\{e^{2}, e^{-2}\}$ .

STEP 1

Assumptions
1. We are given the equation $8|\ln x| - 16 = 0$ .
2. We need to solve for $x$ in terms of the natural number $e$ .
3. The absolute value function $|y|$ splits the equation into two cases, $y$ and $-y$ .
4. The natural logarithm function $\ln x$ is defined for $x > 0$ .

STEP 2

Isolate the absolute value expression by adding $16$ to both sides of the equation.
$8|\ln x| - 16 + 16 = 0 + 16$

STEP 3

Simplify the equation.
$8|\ln x| = 16$

STEP 4

Divide both sides of the equation by $8$ to solve for $|\ln x|$ .
$\frac{8|\ln x|}{8} = \frac{16}{8}$

STEP 5

Simplify the equation further.
$|\ln x| = 2$

STEP 6

Now we need to consider the two cases for the absolute value.
Case 1: $\ln x = 2$
Case 2: $\ln x = -2$

STEP 7

Solve for $x$ in Case 1 by exponentiating both sides of the equation to eliminate the natural logarithm.
$e^{\ln x} = e^2$

STEP 8

Simplify using the property $e^{\ln a} = a$ .
$x = e^2$

STEP 9

Solve for $x$ in Case 2 by exponentiating both sides of the equation to eliminate the natural logarithm.
$e^{\ln x} = e^{-2}$

STEP 10

Simplify using the property $e^{\ln a} = a$ .
$x = e^{-2}$

STEP 11

Combine the solutions from Case 1 and Case 2.
The solution set is $\{e^2, e^{-2}\}$ .

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Solve the equation 8∣ln⁡x∣−16=08|\ln x|-16=08∣lnx∣−16=0. The solution set is {e2,e−2}\{e^{2}, e^{-2}\}{e2,e−2}.

STEP 1

STEP 2

Isolate the absolute value expression by adding 161616 to both sides of the equation.8∣ln⁡x∣−16+16=0+168|\ln x| - 16 + 16 = 0 + 168∣lnx∣−16+16=0+16

STEP 3

Simplify the equation.8∣ln⁡x∣=168|\ln x| = 168∣lnx∣=16

STEP 4

Divide both sides of the equation by 888 to solve for ∣ln⁡x∣|\ln x|∣lnx∣.8∣ln⁡x∣8=168\frac{8|\ln x|}{8} = \frac{16}{8}88∣lnx∣​=816​

STEP 5

Simplify the equation further.∣ln⁡x∣=2|\ln x| = 2∣lnx∣=2

STEP 6

Now we need to consider the two cases for the absolute value.Case 1: ln⁡x=2\ln x = 2lnx=2Case 2: ln⁡x=−2\ln x = -2lnx=−2

STEP 7

Solve for xxx in Case 1 by exponentiating both sides of the equation to eliminate the natural logarithm.eln⁡x=e2e^{\ln x} = e^2elnx=e2

STEP 8

Simplify using the property eln⁡a=ae^{\ln a} = aelna=a.x=e2x = e^2x=e2

STEP 9

Solve for xxx in Case 2 by exponentiating both sides of the equation to eliminate the natural logarithm.eln⁡x=e−2e^{\ln x} = e^{-2}elnx=e−2

STEP 10

Simplify using the property eln⁡a=ae^{\ln a} = aelna=a.x=e−2x = e^{-2}x=e−2

STEP 11

Combine the solutions from Case 1 and Case 2.The solution set is {e2,e−2}\{e^2, e^{-2}\}{e2,e−2}.

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Solve the equation $8|\ln x|-16=0$ . The solution set is $\{e^{2}, e^{-2}\}$ .

Isolate the absolute value expression by adding $16$ to both sides of the equation.
$8|\ln x| - 16 + 16 = 0 + 16$

Simplify the equation.
$8|\ln x| = 16$

Divide both sides of the equation by $8$ to solve for $|\ln x|$ .
$\frac{8|\ln x|}{8} = \frac{16}{8}$

Simplify the equation further.
$|\ln x| = 2$

Now we need to consider the two cases for the absolute value.
Case 1: $\ln x = 2$
Case 2: $\ln x = -2$

Solve for $x$ in Case 1 by exponentiating both sides of the equation to eliminate the natural logarithm.
$e^{\ln x} = e^2$

Simplify using the property $e^{\ln a} = a$ .
$x = e^2$

Solve for $x$ in Case 2 by exponentiating both sides of the equation to eliminate the natural logarithm.
$e^{\ln x} = e^{-2}$

Simplify using the property $e^{\ln a} = a$ .
$x = e^{-2}$

Combine the solutions from Case 1 and Case 2.
The solution set is $\{e^2, e^{-2}\}$ .