How can i solve log 5? Please suggest step by step solution.?

Answer 1

I assume by Log 5 you are referring to Log(10) 5.

First consider this: -

If:-
1 =10^0 (here the ^ symbol means to the power of)
10 = 10^1
100 = 10^2
1000 = 10^3
.
.
.
1000000 = 10^6

Then the powers are the Log(10) values: -

Thus 2 = Log(10)(100) and 6 = Log(10)(1000000) etc..,

Hence, if Log(10)(10.0) = 1 then the Log(10)(5) must be less than 1.

You can use a calculator to find Log(10) 5 = 0.698970004

or you can solve the equation:-

5 = 10^x

where 'x' is the log(10) of 5.

This can be done with a calculator and some refined guesses
on a starting value of x>0.5 and x<0.7.

In more advanced mathematics there exist equations for calculating the value of the log.

However, if you meant Log(e)(5) or natural logs then the natural number e=2.718281828 replaces '10' and the same approach may be used.

I hope this helps.

Answer 2

I'm going to assume you want the natural log of 5.
Start with the geometric series
1/(1+x) = 1 - x + x² - x³ + ... ,
which is good for |x| <1
Now integrate it term by term to get,
ln(1+x) = x - x²/2 + x³/3 + ... , which is also good
for |x| <1.
Now you may ask: "But I want log 5, which would
require x = 4 and this series is no good at x = 4.
What do I do?"
Well, let's use this series to get log 1/5 instead.
Then log 5 = -log 1/5.
So plug x = -4/5 into the above series and
compute it to as many terms as you need,
then change the sign!
This is especially nice if you have a calculator
without a log button.
Only problem here is that this series converges
rather slowly so you may have to use many
terms to get the accuracy you want.
I played with this using PARI
and had to take 26 terms to get the first 3
decimal places, 1.609, correct.

Answer 3

You can estimate it without a calculator.

What is 10(^1/2)? That is, sqrt(10): answer 3.14 or so.
What is 10(^1/3)? That is, cube root of 10. We don't know,
but we do know the cube root of 8 is 2.
So, we'd guess a little under 2.2, say 2.15.
We had better square that, a bit over 4.

What does that tell you about log 5?

We have produced 3.14 and we have produced a bit over 4.

We are getting closer: log 5 is closer to 2/3 than it is to 1/2.
[Think about it].

I would say log 5 is about 0.7.

[All right, you were lost in space, without a calculator and you had to know]
I

Answer 4

log(5)=x (assume base 10)
10^x = 5
You'll have to find an x such that when raised to power 10, it produces 5.
If you cannot use a calculator, you can only solve it by using a computer and numerical methods.

Answer 5

a) Find calculator
b) locate log button
c) enter "log(5)"
d) press enter
e) Read result
f) re-insert brain

Answer 6

log 5 = 0.699