Logarithms, Explained - Steve Kelly

Prevoditelj: Irma Komljenović Recezent: Ivan Stamenković

How does the difference between

Kako razlika između

0.0000000398 i

point 0000000398 and

point 00000000398

0.00000000398

cause one to have red eyes after swimming?

uzrokuje crvenilo očiju nakon plivanja?

To answer this, we first need a way of

Da bismo odgovorili na to, moramo naći način

dealing with rather small numbers,

rukovanja iznimno malim brojevima.

ili, u nekim slučajevima, iznimno velikim brojevima.

or in some cases extremely large numbers.

This leads us to the concept of logarithms.

To nas dovodi do koncepta logaritama.

Well, what are logarithms?

Što su to logaritmi?

Let's take the base number, b,

Uzmimo bazu b

and raise it to a power, p,

i potencirajmo ju brojem p,

like 2 to the 3rd power,

kao 2 na treću,

and have it equal a number n.

a neka to bude jednako broju n.

We get an exponential equation: b raised to the p power, equals n.

Dobivamo eksponencijalnu jednadžbu: b na p jednako je n.

In our example, that'd be 2 raised

U našem primjeru, to bi bilo 2 na treću jednako je 8.

to the 3rd power, equals 8.

The exponent p is said to be

Za eksponent p kažemo

the logarithm of the number n.

da je logaritam broja n.

Most of the time this would be written:

Većinu vremena ovo bi bilo napisano kao:

"log, base b, of a number equals p, the power."

"logaritam po bazi b od nekog broja jednak je p, potenciji".

This is starting to sound a bit confusing with all the variables,

Ovo zvuči pomalo komplicirano zbog svih tih varijabli

so let's show this with an example.

pa ćemo to prikazati primjerom.

What is the value of

Koja je vrijednost

log base 10 of 10,000?

logaritma po bazi 10 od 10,000?

The same question could be asked using exponents:

Isto bi se pitanje moglo postaviti koristeći eksponente:

"10 raised to what power is 10,000?"

"10 na koju potenciju je 10,000?"

Well, 10 to the 4th is 10,000.

Pa, 10 na četvrtu je 10,000.

So, log base 10 of 10,000

Dakle, log po bazi 10 od 10,000

must equal 4.

mora biti jednak 4.

This example can also be completed very simply on a scientific calculator.

Ovaj se primjer može jednostavno završiti na znanstvenom kalkulatoru.

Log base 10 is used so frequently

Log po bazi 10 u znanosti se koristi toliko često

in the sciences

that it has the honor of having its own button on most calculators.

da ima čast imati vlastitu tipku na većini kalkulatora.

If the calculator will figure out logs for me,

Ako će kalkulator izračunati logaritme umjesto mene,

why study them?

zašto da učim o njima?

Just a quick reminder:

Brzi podsjetnik:

the log button only computes logarithms of base 10.

tipka za logaritam računa samo logaritme po bazi 10.

What if you want to go into computer science

Što ako se želiš baviti računarstvom

and need to understand base 2?

i trebaš razumjeti bazu 2?

So what is log base 2 of 64?

Koji je log po bazi 2 od 64?

In other words, 2 raised to what power is 64?

Drugim riječima, 2 na koju potenciju je 64?

Well, use your fingers. 2, 4, 8, 16, 32, 64.

Pa, upotrijebi svoje prste. 2, 4, 8, 16, 32, 64.

So log base 2 of 64 must equal 6.

Log po bazi 2 od 64 jednak je 6.

So what does this have to do with my eyes turning red

Kakve to ima veze sa crvenilom mojih očiju

in some swimming pools

u nekim bazenima,

and not others?

a u drugima ne?

Well, it leads us into an interesting

Pa, dovodi nas do zanimljive

use of logarithms in chemistry:

upotrebe logaritama u kemiji:

finding the pH of water samples.

pronalaženja pH uzoraka vode.

pH tells us how acidic or basic a sample is,

pH nam kaže koliko je kiseo ili lužnat uzorak,

and can be calculated with the formula:

a može ju se izračunati formulom:

pH equals negative log base 10 of the hydrogen ion concentration, or H plus.

pH jednak je negativnom log po bazi 10 od koncentracije vodikovih iona ili H+.

We can find the pH of water samples

Možemo izračunati pH uzoraka vode

with hydrogen ion concentration of point 0000000398

sa koncentracijom vodikovih iona 0.0000000398

and point 00000000398

i 0. 00000000398

quickly on a calculator.

brzinski na kalkulatoru.

Punch: negative log of each of those numbers,

Ukucaj: negativni log svakog od tih brojeva

and you'll see the pH's are 7.4 and 8.4.

i vidjet ćeš da pH iznosi 7.4 i 8.4.

Since the tears in our eyes have a pH of about 7.4,

S obzirom da je pH naših suza oko 7.4,

the H plus concentration of .0000000398

koncentracija H+ koja iznosi 0.0000000398

will feel nice on your eyes,

bit će ugodna našim očima,

but the pH of 8.4 will make you feel itchy and red.

no zbog pH koji iznosi 8.4 oči će ti biti svrbljive i crvene.

It's easy to remember logarithms "log base b of some number n equals p"

Lako je zapamtiti logaritme "log po bazi b nekoga broja n jednak je p"

by repeating: "The base raised to what power equals the number?"

ponavljanjem: "Baza potencirana čime jednaka je broju?"

"The BASE raised to what POWER equals the NUMBER?"

"BAZA POTENCIRANA čime jednaka je BROJU?"

So now we know logarithms are very powerful

Sada znamo da su logaritmi moćni

when dealing with extremely small or large numbers.

u rukovanju s iznimno malim ili velikim brojevima.

Logarithms can even be used

Logaritme čak možemo koristiti

instead of eyedrops after swimming.

umjesto kapi za oči poslije plivanja.