<P> The first step in approximating the common log is to put the number given in scientific notation . For example, the number 45 in scientific notation is 4.5 x 10 ^ 1, but we will call it ax 10 ^ b . Next, find the log of a, which is between 1 and 10 . Start by finding the log of 4, which is . 60, and then the log of 5, which is . 70 because 4.5 is between these two . Next, and skill at this comes with practice, place a 5 on a logarithmic scale between . 6 and . 7, somewhere around . 653 (NOTE: the actual value of the extra places will always be greater than if it were placed on a regular scale . i.e., you would expect it to go at . 650 because it is halfway, but instead it will be a little larger, in this case . 653) Once you have obtained the log of a, simply add b to it to get the approximation of the common log . In this case, a + b = . 653 + 1 = 1.653 . The actual value of log (45) ~ 1.65321 . </P> <P> The same process applies for numbers between 0 and 1 . For example, 0.045 would be written as 4.5 × 10 . The only difference is that b is now negative, so when adding you are really subtracting . This would yield the result 0.653 − 2, or − 1.347 . </P> <P> Natural exponents are used in many important expressions in modern science and engineering, with applications not limited to quantum mechanics, thermodynamics and signal communications . Using the laws of Natural exponents, memorization of the approximations below, and combination with other mental calculation methods, create a powerful and elegant means for changing complicated problems in the physical sciences into simple sums and products . The laws of Natural exponents (Exponentiation) are: </P> <Dl> <Dd> e xe = e and e = 1 / e and also </Dd> </Dl>

What is the first common multiple of 2 3 4 5 6 and 10