- WE WANT TO EVALUATE 
THE COMMON LOGARITHMS
WITHOUT THE USE 
OF A CALCULATOR.
IN THE FIRST EXAMPLE 
WE HAVE COMMON LOG 10,000.
REMEMBER WHEN A LOG IS WRITTEN 
THIS WAY WITH NO BASE
WE KNOW THAT IT'S COMMON LOG 
OR LOG BASE 10.
AND TO EVALUATE THIS COMMON 
LOG, WE'LL SET IT EQUAL TO X,
WRITE IT AS AN EXPONENTIAL 
EQUATION,
AND THEN ONCE WE FIND 
THE VALUE OF X
WE KNOW THE VALUE 
OF THIS LOGARITHM.
TO CONVERT A LOG EQUATION 
TO AN EXPONENTIAL EQUATION,
REMEMBER B IS THE BASE, 
A IS THE EXPONENT,
AND N IS THE NUMBER.
AN EASY WAY TO DO THIS 
IS TO START WITH THE BASE,
WORK AROUND THE EQUAL SIGN TO 
FORM THE EXPONENTIAL EQUATION.
SO HERE WE HAVE 10 RAISED TO 
THE POWER OF X MUST = 10,000.
SO 10 TO THE POWER 
OF X = 10,000.
NOW, THE REASON WE CAN DO 
THESE WITHOUT OUR CALCULATOR
IS BECAUSE WE CAN WRITE 10,000
AS 10 RAISED TO THE SOME POWER 
10,000 IS = TO 100 x 100,
AND 100 IS = TO 10 x 10,
SO 10,000 IS = TO 10 
TO THE FOURTH.
SO WE HAVE 10 TO THE X = 10 
TO THE FOURTH.
A QUICK EASY WAY TO DETERMINE 
THE EXPONENT HERE
IS TO COUNT THE NUMBER 
OF ZEROS.
IF WE HAVE 1 
FOLLOWED BY FOUR 0's
THAT'S = TO 10 TO THE FOURTH.
SO 10 TO THE X = 10 TO THE 
FOURTH MEANS THESE ARE EQUAL
AND THE BASES ARE THE SAME, 
THEREFORE, X MUST = 4,
AND THEREFORE, LOG 10,000 = 4.
LET'S LOOK AT AN EXAMPLE NOW
WHERE THE NUMBER 
IS A FRACTION.
AGAIN, WE HAVE COMMON LOGS,
WE NEED TO RECOGNIZE 
THAT THE BASE IS 10,
WE'LL SET THIS EQUAL TO Y.
IF WE DETERMINE THE VALUE 
OF Y,
WE KNOW THE VALUE 
OF THIS LOGARITHM.
SO NOW I'LL WRITE THIS 
AS EXPONENTIAL EQUATION,
SO THE BASE IS 10, 
THE EXPONENT IS Y,
AND IT'S = TO 1/1,000.
SO WE HAVE 10 TO THE POWER 
OF Y = 1/1,000.
NOTICE THAT 1,000 
HAS THREE 0's,
SO 1,000 IS = TO 10 
TO THE THIRD.
SO WE HAVE 10 TO THE Y = 1/10 
TO THE THIRD.
AND NOW WE CAN USE 
OUR EXPONENT PROPERTIES
TO REWRITE THIS.
IF WE MOVE THIS 
ACROSS THE FRACTION BAR
OR MOVE THIS UP 
TO THE NUMERATOR,
IT'S GOING TO CHANGE THE SIGN 
OF THE EXPONENT.
SO THIS WOULD GIVE US 10 
TO THE Y = 10 TO THE -3.
AGAIN, IF THESE ARE EQUAL 
AND THE BASES ARE THE SAME,
THEN THE EXPONENTS 
MUST BE EQUAL.
THEREFORE, Y IS = TO -3.
THEN IF Y IS = TO -3
THEN THE ORIGINAL COMMON 
LOGARITHM IS = TO -3.
MEANING THE COMMON LOG 
OF 1/1,000 IS = TO -3.
NEXT WE'LL LOOK 
AT TWO EXAMPLES
OF EVALUATING COMMON LOGS
WHERE WE CANNOT WRITE 
THE NUMBER AS 10 TO A POWER.
AND THEREFORE, WE'LL HAVE 
TO USE THE CALCULATOR.
I HOPE YOU FOUND THIS HELPFUL.
