## Thursday, December 10, 2009

### LOGARITHMS_ 6.2

The inverse of a function is what we now know as logarithm. Logarithm, which many people fail to understand fully, is simply the power or exponent (y) to which the given base must be raised to in order to produce a known value.
For example,
When solving for 10 ^y= 100, we know that in order to get 100, 10 must be multiplied to itself two times; 10*10 = 100, in other words, 10^ 2 = 100.
But when solving for an equation such as 10^x=32, trial and error could be quite exhausting to work with and so we use John Napier’s invention of the LOGARITHM, which simply states that if x=b^y, then y=logb(x).
Using logarithm, we can now state the exponential form, 10^2=100, in the logarithmic form, log(10)100=2 and the exponential form, 10^x=32, in the logarithmic form, log10(32)=x.
Often, however, when speaking of a base 10 in logarithm, the base value will be written as log, and will not be stated as it is understood to mean the same as log10. Such logarithms with a base 10 are known as common logarithms.

Below is a graph of a y=10^x function show in red and its inverse shown in blue. As you can see there is a strong relationship with y=1.5 as shown with the black line, and x=10^y, also known as y=log(x), as they intersect at a point which represent the x-value.

When we think of an inverse function we think of the x and y value switching places, similarly, when using log, an easier way to understanding the concept at the very basic level before growing further into this unit is to see it as the same; a switch in the positions of the x and y value. Such as the inverse of y=b^x which is x=b^y where you can see the x and y have switched positions, using the logarithm, we can similarly see that when we simply switch the positions of the inverse function we have, y=logbx, where b represents the base value.

EXAMPLE 1:
The exponential form, 4^3=64 can be written in the logarithmic form as
Log464=3 (recognise how the values highlighted in blue representing x and green representing y switch places)
EXAMPLE 2:
The logarithmic form, log2(64) can be written in the exponential form as
2*x=64
àUsing previous experience and knowledge, we know that 2^6 = 64, so evaluating the logarithm we get...
X=6

As we have seen, sometimes the exponents are not very easy to solve for, and so we turn to our calculators. So far, we have only learned to solve for base 10.
When using our calculators to solve for base 10 functions that we cannot solve on our own, we simply press LOG on our calculators, and type in the equation as we see it; so for instance if we were solving for log 0.001 (remember, this is like saying log10(0.001)), we simply PRESS log, and then type in 0.001; the calculator assumes that we are working with base 10 in such a scenario.
- Michelle Joseph