# why does log 10 100,000 equal to 5? Why must log 10 5 be a number between 0 and 1?

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• Definition

The log of a number to a given base is the index of the power to which the base must be raised in order to give the number.

Examples

1000   = 10³

log 10 (1000)   = 3

16= 2⁴

log 2 (16)   = 4

Thus

100 000   = 10^5

log 10 (100 000)   =  5

• log 10 100,000

log 10 (10)^5

= 5

10- base

100,000- final product

5- power

Logarithms (without the given base) are done in 10s

• Knowing some basic facts, log 1=0 and log10= 1

we can write

log100= log10^2 = 2 log10=2*1=2

log1000= log10^3= 3log10 =3*1=3

log10000=log10^4=4 log10=4*1=4

log100000= log10^5= 5log10=5*1=5

and so on....

basically number of zeros tells the value of log.

We know that log1=0 and log10=1 so  log values of all the numbers lying between 1 and 10 would lie between 0 and 1.

Based on this  we can say that log5 would also be a number between 0 and 1

https://celestialtutors.com/topic/exponential-and-...

• Log base 10 of a number means 10 raised to what power will give you that number.

10^5 = 100,000

So that exponent, 5, is the Log of 100,000

• log(base 10)100000 = 5 because 10^5 = 100000

10^0 = 1

10^1 = 10

1 < 5 < 10

Therefore, 0 < log(base 10)5 < 1.

• rule 1: log[base a](b^c) = c log[base a](b)

rule 2: log[base a](a) = 1

rule 3: log without base written is assumed to be base 10

log(100,000) = log(10^5) = 5log(10) = 5

• 100,000 = 10^5

100000 = 10^(log(100000))

so, log(100000) = 5

For the other part:

10^0 = 1 ==> log(1) = 0

10^1 = 10 ==> log(10) = 1

1 < 5 < 10

so log(5) must be between 0 and 1

• Because 10^5 = 100,000

• Anonymous
8 months ago

log(10) 100,000 = log(10)10^5 = 5

log(10) 5 would be found in the interval log(10)1<log(10)5<log(10)10 or log(10)10^0 <log(10)5 <log(10)10^1 or 0<log(10)5<1

• Why does log₁₀(100000) equal 5?

When the base of the log and the base of the number inside are the same, they cancel each other out leaving only the exponent.  So putting  your number into a power of 10 we get:

log₁₀(100000)

log₁₀(10⁵)

The 10's cancel out leaving only:

5

-------------

Why must log₁₀(5) be a number between 0-1?

Because the log of any base of 1 is 0, because the 0th power of any non-zero base is 1.  Using the example above we can say:

log₁₀(10⁰) = 0

log₁₀(1) = 0

So the log of 1 from any base is 0.

To get the log to equal 1, the value inside needs to be 10:

log₁₀(10¹) = 1

Since 5 is between 10⁰ and 10¹, the log of 5 is between 0 and 1.