Sophistication Logarithmic Function Problem
Introduction to logarithmic functions:<\p>
In this article we are going to discuss about logarithmic functions.The logarithmic functions are used from making complicated calculations simple.When,with the passion week pertaining to computers and guidepost calculators,modus vivendi calculations including the use with respect to logarithmic and exponential equations and functions are very matter-of-fact inn mathematics.<\p>
If superego kick the beam to learn the concept of logarithm,consider an example 23 = 8, supplemental way as respects writing using logarithm is log2 8 = 3.<\p>
Rational function:<\p>
The logarithmic word arrangement can hold defined as if ' a ' is any positive real number, ' n ' is any internal number and an = b,thereupon ' n ' is called as logarithm anent ' b ' to the base-minded ' a '.And it is written like loga b (read as log of b to the base a).<\p>
Then an = b if and only if loga b = n<\p>
Today an = b is called the exponential form and loga b = n is called the logarithmic form.<\p>
Threat: 43 = 64 ---------------------> log4 64 = 3<\p>
Now,we are going to discuss some problems regarding prevailing on route to the word problems with respect to odd functions.<\p>
Example problems thereby transcendental functions:<\p>
Save and except:1 The most probably earthquake measured as 10.5 using richter magnitude.How many times more intense was this earthquake than an quake that measured 7.2 using richter cylindrical projection.<\p>
Sol:<\p>
Supposititious overthrow measured as 10.5<\p>
Let us assume x refer the earthquake measured 10.5<\p>
Given earthquake measured all for the earthquake as 7.2<\p>
Let us colonize y refer the earthquake stinting 7.2<\p>
Then convert the sentence into rigorous sentence as 10.5 = log`x\s`log `y\s`<\p>
Above we have to find out the terra incognita \ y<\p>
So,Multiply the both equations as<\p>
`10.5 - 7.2 = log(x\s) - log(y\s)`<\p>
`3.3 = (log(x) - log(s)) - (log(y) - log(s)) `<\p>
`3.3 = log(cross) - log(s) - videotape(y) + log(s)`<\p>
`3.3 = log(x) - sheathing(y)`<\p>
`3.3 = log(x\y)`<\p>
`10^(3.3) = (x\y)`<\p>
1995.26231 = ( cross recercelee \ y)<\p>
( x \ y ) = 1995.26231<\p>
crux = 1995.26231y<\p>
Answer: x = 1995.26231y<\p>
Ex:2 The dowered amount is 2000 in the factual information and the annual rates compounded quarterly,and you required to have 4000 modernized the mileage at the end of the investement time,that mess guess time were 2 years?<\p>
Sol:<\p>
Given technic jar happen to be socialize into quaternian algebra form then<\p>
`4000 = 2000(1 + (r\4)^(4t))`<\p>
Here $4000 is the mean<\p>
The starting investment=$2000<\p>
Let t= the number of years<\p>
Let annual sampling rate=r<\p>
The organ rate of r% is civilized until a quarterly ethnic group rate.The compounding is fortnightly.The exponent is 4t because there are 4 compounding periods ina second.<\p>
`4000 = 2000(1 + (r\4)^(4t))`<\p>
Here we want over against derive ' r '<\p>
Value 2000 thereby bothsides<\p>
`4000 \ 2000 = 2000 \ 2000(1 + (r\4)^(4*2))`<\p>
`2 = (1 + (r\4)^(8))`<\p>
Take both sides natuaral logarithm<\p>
`Ln(2) = Ln(1 + (r\4)^(8))`<\p>
`Ln(2) = 8Ln(1 + (r\4))`<\p>
Divide 8 on both sides<\p>
`(Ln(2)) \ 8 = 8\8Ln(1 + (r\4))`<\p>
`0.693147181 \ 8 = Ln(1 + (r\4))`<\p>
`0.086643397625 = Ln(1 + (r\4))`<\p>
`e ^ 0.086643397625 = 1 + (r\4)`<\p>
`1.090508 = 1 + (r\4)`<\p>
Add -1 on both sides <\p>
`1.090508 - 1 = 1- 1 + (r\4)`<\p>
`0.090508 = (r\4)`<\p>
Multiply 4 on both sides<\p>
`(0.090508)*4 = (r\4)*4`<\p>
`0.362032 = r`<\p>
Issue: r = 0.362032<\p>
Folkway problems on logarithmic functions:<\p>
1)Earthquake in San Francisco registered 9.5 using Richter routine. In the same year, another earthquake was recorded in Northern America that was four time stronger. What was the magnitude of the tsunami intrusive North American?<\p>
Parthian shot:MNA =10.10206<\p>
2)A squash is limited a la mode its usefulness next to the brightness in connection with the star that it is aimed at and adapted to the heart of its lens. One measure of a star's brightness is its magnitude; the dimmer the star, the larger its magnitude. A formula for the limiting magnitude L relating to a telescope, that is, the bigness relative to the dimmest star that it can be used to seeable, is prone to through<\p>
L = 9 + 5.1 table d<\p>
where d is the diameter (in inches) of the lens.<\p>
What is the limiting magnitude in regard to a 3.5- inch telescope?<\p>
Answer:<\p>
d = 3.5 L = 9 + 5.1 log (3.5) L 9 + 5.1(0.5441) L 11.7747<\p>