Math  /  Algebra

QuestionIdentify the solution set of 6lne=eln2x6 \ln e=e^{\ln 2 x} \{2\} \{3\} \{6\}

Studdy Solution

STEP 1

What is this asking? We need to find the value of xx that makes the equation 6lne=eln2x6 \ln e = e^{\ln 2x} true. Watch out! Don't forget those logarithm rules! ee and ln\ln are best buds, and they like to simplify things when they're together.

STEP 2

1. Simplify the left side
2. Simplify the right side
3. Solve for *x*

STEP 3

We've got 6lne6 \ln e on the left.
Remember, lne\ln e is the same as asking "*ee to what power equals ee*?".
Since e1=ee^1 = e, we know lne=1\ln e = 1!

STEP 4

So, 6lne6 \ln e simplifies to 61=66 \cdot 1 = 6.
Our equation now looks like 6=eln2x6 = e^{\ln 2x}.

STEP 5

Now, let's look at the right side: eln2xe^{\ln 2x}.
This is asking "*ee to the power of *what* equals 2x2x*?".
But ln2x\ln 2x *is* asking that!

STEP 6

So, eln2xe^{\ln 2x} simplifies to just 2x2x.
Our equation is now 6=2x6 = 2x.
Much cleaner, right?

STEP 7

We're almost there!
We have 6=2x6 = 2x.
To get xx by itself, we need to divide both sides of the equation by 2\textbf{2}.

STEP 8

Dividing both sides by 2\textbf{2} gives us 62=2x2\frac{6}{2} = \frac{2x}{2}.
This simplifies to 3=x3 = x, or x=3x = 3.
Boom!

STEP 9

The solution set is \{3\}.

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