Brooke Kazama Post 4

Logarithmic Functions:

A Logarithmic Function is a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.

                There are several applications that require the use of logarithmic functions, some of these applications include carbon dating, population growth, and investment interest.  

Essentially,

if X is a positive number, log X is the exponent of 10 that gives X

In other words,

                If       Y = log X         then       10^Y  = X

The best way to think about logs are in terms of exponents. This is a good way to be able to visualize and answer logarithmic equations.

Here is an example:

Log 100 = 2

                Looking at the original equation and knowing that log X = Y is the same as 10^Y = X then the exponential equation would be

10²

                To check, you can solve. And it turns out

10² = 100  

Now, taking a look at logarithmic functions and exponential functions, it turns out that these two functions are inverses of each other

Log (10^X) = X                              and if X > 0        10^{log X} = X

Now there are also properties that will help solve logarithmic equations:

1.       Log 1 = 0      and      log 10 = 1

2.       Log (10^X) = X   ,  And if X > 0     10^{log X} = X

3.       a and b are both positive numbers and t can be any value  

i.                      log (ab)= log a + log b

ii.                   Log (a/b) = log a – log b

iii.                  Log (b^t) = t log b   

Along with logarithm functions, there are also logs called “natural logarithms”, essentially the only difference between the two is the fact that instead of having Log₁₀ , the natural log looks like ln with a base of “e”   

Thus,  for X > 0

                Ln X = Y      means that    e^Y = X

The natural log is very similar to the logarithmic function, here are the properties of the natural log

1.       Ln 1 = 0    and     ln e = 1

2.       The function e^X and ln X are inverses of each other

Ln(e^X) = X  for all values of X

e^{log X} = X     for all values of X > 0

3.       a and b are positive and t are all values

i.                     ln (ab) = ln a + ln b

ii.                   ln (a/b) = ln a – ln b

iii.                  ln (b^t) = t ln b

but there are always some misconceptions concerning logs, here are some mistakes that you should be on the lookout for!   

Log (a + b) not the same as log a + log b

Log (a – b) not the same as log a – log b

Log (ab) not the same as (log a) (log b)

Log (a/b) is not the same as log a / log b

Log (1 / a) is not the same as 1/ log a