Applications+of+Logarithm

=__Application 1 (mathematical)__=

We used 'e' here

a bacteria population that grows at 5% rate per hour continuously (in other words, so many cells that the "birth" of a new cell can be considered occurring at all times). You can then determine how long it will take to double the population:

y=y0*e^(0.05t)

// where y=current population, // // y0=initial population, // // t=point in time, // // e=2.71828.... //

So we want to know when the population doubles, when its current population will be double its initial population:

2y0=y0*e^(0.05t)

2=e^(0.05t)

ln(2) = ln[e^(0.05t)] ln(2)=0.05t*ln(e)

but ln(e)=1

ln(2)=0.05t

so t = ln(2)/0.05 = 20ln(2) = 13.8629 hours

The continuous aspect of this problem tells you that you need to use the exponential function.

people use devices or scales that are based on logarithms. Every time that you hear about an earthquake on the news, the earthquake is described in reference to a scale based on logarithms. Earthquakes happen daily all over the world. logarithms help save many people that are affected by earthquakes. To be more specific, the ritcher scale uses the logarithms!
 * __Application 2 ( In words ) __**

for example : __ The quake, estimated at 5.4 magnitude (reduced from an initial estimate of 5.8), was centered 35 east of downtown Los Angeles in Chino Hills, just south of Pomona in San Bernardino county. It was felt as far east as Las Vegas and as far south as San Diego. ____ My first reaction, a question: how much of a difference was there in terms of the seismic energy released at the epicenter of the estimated earthquake versus the actual earthquake? How off was the esimate? I know that the Richter Scale is logarithmic, so the answer would be: __



__ as you can see, this is used to calculate the magnitude. __

== Logarithms are useful in at least two major circumstances: One is where exponential functions are used. Just as division is the inverse of multiplication, a logarithm is the inverse of an exponential function. In physics, a common use is radioactive decay. The amount of radioactive material remaining after some time passes is the product of the initial amount and an exponential function of (-t/t0). To calculate the amount remaining as a function of time requires an exponential function. To go backwards, to calculate the time passed as a function of the amount of radioactive material remaining, requires a logarithmic function. ==

== a definition of logarithms: The "log(N)" to the base "a" is the power "p" to which "a" must be raised to obtain "N". So: a^p = N or a^(log(N)) = N. For example, 100 = 10^2 so the log of 100 to the base 10 is 2. Or in a less familiar example: 243 = 3^5. So 5 is the log(243) to the base 3. ==