Math / Algebra

QuestionSolve: $3 \ln x=\ln 216$ $x=3$ $x=6$ $x=72$

Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that makes $3 \ln x$ equal to $\ln 216$ . Watch out! Remember the logarithm power rule!
Don't accidentally divide by 3 instead of using the power rule.

STEP 2

1. Rewrite the equation
2. Solve for $x$

STEP 3

We have $3 \ln x = \ln 216$ .
Remember the power rule: $n \cdot \ln a = \ln (a^n)$ .
So, we can rewrite the left side of the equation: $3 \ln x = \ln (x^3)$ .
Why are we doing this?
This magical power rule lets us get our variable $x$ out of that logarithm's tight grip!
Now our equation looks like this: $\ln (x^3) = \ln 216$ .
Much better!

STEP 4

Since we have the same logarithm ( $\ln$ ) on both sides of the equation, we can set the arguments equal to each other.
Think of it like this: if two boxes labeled $\ln$ have the same contents, then the stuff inside the boxes must be the same!
So, we get $x^3 = 216$ .

STEP 5

Now, how do we get $x$ all by itself?
We need to undo that cube, and we do that by taking the cube root of both sides.
So, $\sqrt[3]{x^3} = \sqrt[3]{216}$ .
This gives us $x = 6$ because $6 \cdot 6 \cdot 6 = 216$ .
Boom!

STEP 6

The solution to the equation $3 \ln x = \ln 216$ is $x = 6$ .

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QuestionSolve: $3 \ln x=\ln 216$ $x=3$ $x=6$ $x=72$

Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that makes $3 \ln x$ equal to $\ln 216$ . Watch out! Remember the logarithm power rule!
Don't accidentally divide by 3 instead of using the power rule.

STEP 2

1. Rewrite the equation
2. Solve for $x$

STEP 3

STEP 4

Since we have the same logarithm ( $\ln$ ) on both sides of the equation, we can set the arguments equal to each other.
Think of it like this: if two boxes labeled $\ln$ have the same contents, then the stuff inside the boxes must be the same!
So, we get $x^3 = 216$ .

STEP 5

Now, how do we get $x$ all by itself?
We need to undo that cube, and we do that by taking the cube root of both sides.
So, $\sqrt[3]{x^3} = \sqrt[3]{216}$ .
This gives us $x = 6$ because $6 \cdot 6 \cdot 6 = 216$ .
Boom!

STEP 6

The solution to the equation $3 \ln x = \ln 216$ is $x = 6$ .

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QuestionSolve: 3ln⁡x=ln⁡2163 \ln x=\ln 2163lnx=ln216 x=3x=3x=3 x=6x=6x=6 x=72x=72x=72

Studdy Solution

STEP 1

What is this asking? We need to find the value of xxx that makes 3ln⁡x3 \ln x3lnx equal to ln⁡216\ln 216ln216. Watch out! Remember the logarithm power rule!Don't accidentally divide by 3 instead of using the power rule.

STEP 2

1. Rewrite the equation2. Solve for xxx

STEP 3

STEP 4

Since we have the **same logarithm** (ln⁡\lnln) on both sides of the equation, we can set the **arguments** equal to each other.Think of it like this: if two boxes labeled ln⁡\lnln have the same contents, then the stuff inside the boxes must be the same!So, we get x3=216x^3 = 216x3=216.

STEP 5

Now, how do we get xxx all by itself?We need to undo that cube, and we do that by taking the **cube root** of both sides.So, x33=2163\sqrt[3]{x^3} = \sqrt[3]{216}3x3​=3216​.This gives us x=6x = 6x=6 because 6⋅6⋅6=2166 \cdot 6 \cdot 6 = 2166⋅6⋅6=216.Boom!

STEP 6

The solution to the equation 3ln⁡x=ln⁡2163 \ln x = \ln 2163lnx=ln216 is x=6x = 6x=6.

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QuestionSolve: $3 \ln x=\ln 216$ $x=3$ $x=6$ $x=72$

What is this asking? We need to find the value of $x$ that makes $3 \ln x$ equal to $\ln 216$ . Watch out! Remember the logarithm power rule!
Don't accidentally divide by 3 instead of using the power rule.

1. Rewrite the equation
2. Solve for $x$

Since we have the same logarithm ( $\ln$ ) on both sides of the equation, we can set the arguments equal to each other.
Think of it like this: if two boxes labeled $\ln$ have the same contents, then the stuff inside the boxes must be the same!
So, we get $x^3 = 216$ .

Now, how do we get $x$ all by itself?
We need to undo that cube, and we do that by taking the cube root of both sides.
So, $\sqrt[3]{x^3} = \sqrt[3]{216}$ .
This gives us $x = 6$ because $6 \cdot 6 \cdot 6 = 216$ .
Boom!

The solution to the equation $3 \ln x = \ln 216$ is $x = 6$ .