What is the value of x when the unknown number appears as both the base and the exponent?
This is an interesting exponential equation that requires a different approach from ordinary algebra problems.
In this video, we explore how to solve xˣ = 100 step by step using mathematical reasoning.
Topics covered: - Exponential equations - Powers and exponents - Finding the value of an unknown - Solving equations involving variables in the exponent - Mathematical reasoning and problem-solving
Try solving the question yourself before watching the complete explanation!
Whether you're a student preparing for advanced mathematics, revising algebra, or simply someone who enjoys challenging maths puzzles, this question will test your problem-solving skills.
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This looks impossible, but let's do it together and see.
because 1 squared is actually equal to 1 times 1, right
which is 1 and 2 squared is 2 times 2
Which is 2, which is 4, rather.
Then we have 3 to the power of 3.
is equal to 3 times 3 times 3
which is 27.
It's still not up to 100 right now.
if we say 4 to the power of 4
and this will be 4 times 4 times
4 times 4. So this is the same as 16 multiplied
by 16, and that's 16 squared right now. Let's use a
A a yat ki ni ye pure
Sixth month of the year is June.
1 Shuyu Chu Ro Ro
1 times 6 is 6, 6 times 2 is 12, so it's going to be 256.
I have these math strips in this channel,
Just check one of my playlists, Math Strips.
And you're going to see a lot of them.
Like this, so the answer is 256.
so we have 256. Now we are looking for
a number raised to the power of itself
that the answer will give us 100.
And we can see that the answer lies between 3 and 4, right?
So, it means that 3 is less than x.
which is less than 4.
So, it means that our answer is a decimal between 3 and 4.
Now how do we find its value?
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Because we have x to the power of x,
Let us add a line to both sides of this equation.
So, if we draw a line to both sides, we will
x raised to the power of x is equal to 100.
Now, note that line a to the power
A is equal to line A.
This comes down.
to multiply ln a, therefore we can
Now write this equation, this one.
As x ln x equals ln 100
Now, 100 is equal to 10 squared.
Okay, so we can now write this as ln.
x is equal to ln 10 to the power
Now, applying this rule as well,
We're going to write this as 2 ln 10, right?
So, we have x ln x
is equal to 2 ln 10.
Nice.
Now, there is something you need to know.
What is equal to e raised to the power of ln x, right?
Yes, x is equal to e raised to the power of ln x.
And that means that
I can write this for me to erase it, which means that
that I can write this equation as
x is e ln x, so now we have e ln x multiplied
ln x multiplied by ln x equals 2
In 10, okay, 2 ln 10. Don't worry, we're almost
We're almost done with this.
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my old subscribers who have been there all along to support me.
Thank you very much.
Okay, now this is what we have.
Now, we can write this as ln x.
e raised to the power of ln x equals 2 times ln 10.
Now, remember.
The lambda w function, I love that function
so very much the lambda w function it says that
when you take the lambda w function
e to the power of a right that this is equal to a
Now take a look at what we have here—you can see
that we can do something like this, there
so if we take the lambda w function
On both sides we have ln x e ln x
is equal to 2, okay, lambda w function of 2, and then 10, okay
so from here, if you compare this and this
You can see that our a is equal to ln x, right?
simply means that this side will give us what
ln x, since this is equal to a, then this will give us ln x
so we are going to have ln x is equal to
to the double number function.
Off to
Okay, now also recall that x
is equal to e to the power
of ln x, right? So if we take, if we put
raise e to both sides of this equation right
so we are going to have e to the power of ln x
is equal to e to the power of lambda times w function of 2 lambda times 10.
Hmm, amazing.
Now this will give us x, right? Is equal
e to the power of the lambda w function
In 10 now x is equal to with your calculator
Okay, this will give you 3.57
And that's the answer, thank you.
So much for watching, and see you in my next video. Bye.