Math  /  Algebra

Questionlne3=\ln e^{3}=

Studdy Solution

STEP 1

What is this asking? What's the natural logarithm of *e* to the power of 3? Watch out! Don't forget what the natural logarithm actually *means*!

STEP 2

1. Define the natural logarithm
2. Simplify the expression

STEP 3

Alright, let's **break this down**!
The natural logarithm (ln\ln) is a special kind of logarithm.
It's the logarithm with base *e*, where *e* is that special mathematical constant approximately equal to 2.71828.
So, when we see ln(x)\ln(x), we're really asking: *"e* raised to what power gives us *x*?"

STEP 4

In our problem, we have ln(e3)\ln(e^3).
So we're asking: *"e* raised to what power gives us e3e^3?".
Think about it!

STEP 5

Let's write it out explicitly.
We're looking for a value yy such that ey=e3e^y = e^3.

STEP 6

Now, this is where it gets exciting!
Look at both sides of the equation.
They both have the same base, which is *e*!
That makes things super easy.

STEP 7

Since the bases are the same, the exponents *must* be equal for the equation to hold true.
That means y=3y = \textbf{3}!

STEP 8

So, ln(e3)=3\ln(e^3) = \textbf{3}.
Boom!

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