Math Is Fun Forum

paulb203

Member

Registered: 2023-02-24

Posts: 348

Dividing powers of 10

10^6/(2)
=1,000,000/2
=500,000
*
Is there a way of doing this without converting 10^6 to 1,000,000?
For example, in the way we would do;
10^6/(10^2)=10^6-2
10^6-2=10^4

Can I express 2 in terms of 10^y?
2=10/5
10/5= what, in terms of 10^y?
Is it possible to express 10/5 in terms of 10^y?

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Prioritise. Persevere. No pain, no gain.

KerimF

Member

From: Aleppo-Syria

Registered: 2018-08-10

Posts: 257

Re: Dividing powers of 10

This came to mind (though not exactly what you have in mind):
10^6/(2) = 10/2*10^(6-1) = 5*10^5 = 500,000

Last edited by KerimF (2024-10-28 03:38:04)

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Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).
But only a human may have the freedom and ability to oppose his natural robotic nature.
But, by opposing it, such a human becomes no more of this world.

paulb203

Member

Registered: 2023-02-24

Posts: 348

Re: Dividing powers of 10

Thanks, Kerim F

I managed to follow that, although I did need pen and paper

---

Prioritise. Persevere. No pain, no gain.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Dividing powers of 10

We are into the realms of logarithms:

So log is the inverse function of 'to the power'

a is called the base and can be anything. Most common are logs in base 10 and logs in the natural log base, e.  Best to have some calculus to appreciate the latter.

On a (scientific) calculator the natural log base button is marked ln, and the base ten button log.

So back to the question

Can I express 2 in terms of 10^y?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

paulb203

Member

Registered: 2023-02-24

Posts: 348

Re: Dividing powers of 10

Thanks, Bob.

Logarithms are on my to-try-list, but further down the line. Looking forward to them.

I did try out 10^0.3010 and got ≈ 2

Then I tried 10^6-0.3010 and got ≈ 500,000

Q. Should that 6-0.3010 be in brackets?

---

Prioritise. Persevere. No pain, no gain.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Dividing powers of 10

Yes as it's 10 to the whole expression.

I'm of an age where there were no calculators to use for calculations. So logs were taught early on as you can make use of this property:

This converts a multiplication into an addition. Similarly divisions and roots.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

paulb203

Member

Registered: 2023-02-24

Posts: 348

Re: Dividing powers of 10

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Prioritise. Persevere. No pain, no gain.

Thaters

Novice

Registered: 2025-01-14

Posts: 3

Re: Dividing powers of 10

Great question! You're diving into some interesting math here.

To address your query: expressing
2
2 (or
10
5
5
10

) in terms of
1
0
?
10
y
  isn’t straightforward, because 2 isn’t a power of 10. Powers of 10 (
1
0
?
10
y
) are logarithmic in nature and are used to describe numbers that are exact multiples or fractions of 10 raised to some exponent. However,
2
2 can still be expressed mathematically using logarithms.

Here’s how:

Start by representing
2
2 using base 10 logarithms:

2
=
1
0
log
⁡
10
(
2
)
2=10
log
10

(2)

This means
2
2 is equivalent to
1
0
?
10
y
, where
?
=
log
⁡
10
(
2
)
y=log
10

(2).

The value of
log
⁡
10
(
2
)
log
10

(2) is approximately
0.30103
0.30103. So:

2
≈
1
0
0.30103
2≈10
0.30103

Now, if you wanted to work with your original division:

1
0
6
/
2
10
6
/2
Using the representation
2
=
1
0
0.30103
2=10
0.30103
, you could rewrite this as:

1
0
6
/
1
0
0.30103
=
1
0
6
−
0.30103
≈
1
0
5.69897
10
6
/10
0.30103
=10
6−0.30103
≈10
5.69897

This approach avoids explicitly converting
1
0
6
10
6
  to 1,000,000 and maintains everything in terms of powers of 10. The downside is that it introduces logarithms, which may or may not simplify your calculation depending on the context.

In summary, while
2
2 itself can’t be neatly expressed as a simple power of 10, using logarithms allows you to still work within the framework of
1
0
?
10
y
. Let me know if you'd like further clarification!

10^6/(2)
=1,000,000/2
=500,000
*
Is there a way of doing this without converting 10^6 to 1,000,000?
For example, in the way we would do;
10^6/(10^2)=10^6-2
10^6-2=10^4

Can I express 2 in terms of 10^y?
2=10/5
10/5= what, in terms of 10^y?
Is it possible to express 10/5 in terms of 10^y?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Dividing powers of 10

hi Alison,

Something strange has happened with the formating of your post. On my laptop there are too many new line characters so it's hard to work out what you're saying.  Does it look ok to you?

If so, how are you viewing the forum?

Or are you an AI bot?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,441

Website

Re: Dividing powers of 10

hi Alison,

Something strange has happened with the formating of your post. On my laptop there are too many new line characters so it's hard to work out what you're saying.  Does it look ok to you?

If so, how are you viewing the forum?

Or are you an AI bot?

Bob

It's an AI bot, yes. I see that in their first post they'd inserted a link in it and all their posts read as though ChatGPT had constructed them.