Question

Can someone teach me logarithims? MEDAL & FAN

Answer 1

Sure

Answer 2

do you have a specific question you want answered, or just the whole enchilada explained?

Answer 3

The whole thing. All I know so far is that it is in the exponential function form and that it is the inverse function of an exponential function.

Answer 4

ok, that's a good start.

when we speak of a logarithm, it is with the understanding that we have a certain base logarithm in mind. depending on the field, common values of that base assumed rather than expressed might be 10 (common for engineering, science, beginning math books), \(e\approx 2.781828...\) for more advanced math and some science, \(2\) for computer science. In all cases, the logarithm and the base have this relationship:

\[\log_b x=a\]\[b^a=x\]

Answer 5

The class I'm learning this in is cambridge additional math

Answer 6

logarithms have some convenient properties which make them very useful for calculations.

\[\log (u*v) = \log u+ \log v\]\[\log(\frac{u}v)=\log u- \log v\]\[\log(u^v)=v\log u\]

Answer 7

Are each of these used in different cases?

Answer 8

also, \[\log(u) = -\log(\frac{1}{u})\]

Answer 9

let's do some examples.

what is the logarithm to the base 10 of 100? in other words, what power of 10 is equal to 100?

Answer 10

2

Answer 11

right. and the log base 10 (everything will be base 10 from now on, unless I ask otherwise) of 1000?

Answer 12

3

Answer 13

right. sorry, had some browser trouble.

so if we want to multiply \[100*1000\]we can take their logarithms and add them to get the logarithm of the product:\[\log (100*1000) = \log(100)+\log(1000)=2+3=5\]

that means if we take our base, \(b=10\), and raise it to the 5th power, we will have our answer:\[10^5=100000\]and indeed, \(100*1000=100000\)

let's try another:\[\log(\frac{1000}{100})=\log{1000}-\log{100}=3-2=1\]but \[\frac{1000}{100}=10\]\[\log 10 = 1\]so that principle appears to work as well

one more:
\[\log(100^2) = 2\log(100)=2*2=4\]but\[\log(100^2)=\log(100*100)=\log 100+\log 100 = 2+2=4\]and\[\log(100^2)=\log(10000)=4\]

Answer 14

now, how is this useful, I hear you thinking?

well, if you have a table of logarithms, you can convert multiplication and division problems into addition and subtraction problems. and many scientific phenomena are essentially logarithmic in nature, not linear. hearing and vision for example both work on log scales.

old-fashioned slide rules work on the basis of logarithms. the scales are logarithmic, so 0-0.1 takes up much of the scale, 0.1-0.2 takes up somewhat less, and so on, with 0.9-1.0 being the final, smallest portion. To multiply two numbers with a slide rule, you mentally converted them to scientific notation, set the end of the slider opposite the mantissa (fractional part) on the outer scale, then moved the sliding window to the mantissa of the second number, and read off the value on the fixed scale. That combined with the sum of the exponents of the two numbers you multiplied gave your answer. You could chain multiple operations together, and various additional scales offered the ability to do logarithms, trig functions, etc. depending on how fancy a slide rule you bought.

Answer 15

if you like playing with numbers, memorizing the logarithms of a few small prime numbers (2,3,5,7, etc.) will allow you to mentally calculate many logarithms. for example, \[\log 2\approx 0.30103\]\[\log 3\approx 0.477121\]if we needed \(\log 6\) we could find that by \[\log 6 = \log 2*3 = \log 2 + \log 3 \approx 0.30103+0.477121 = 0.778151\]

Answer 16

logarithms work very well for handling quantities that differ by huge amounts. for example, in chemistry, the pH of a solution is the concentration of hydrogen ions, which can go from essentially 0 to numbers with exponents of \(10^{10}\) or more. a linear scale would be much too precise on one end, and much too unwieldy, but a logarithm scale works perfectly. which would you rather deal with, \(3.1*10^{-8}\) or 7.51? Earthquakes are also measured on a logarithmic scale, as is sound pressure (dB).