- WE WANT TO WRITE 
THE EXPONENTIAL EQUATIONS
AS LOGARITHMIC EQUATIONS.
TO DO THIS, 
WE NEED TO UNDERSTAND
THIS LOGARITHMIC EQUATION IN RED
AND THIS EXPONENTIAL EQUATION 
IN RED
MEAN THE SAME THING.
LOG BASE B OF N = "A"
MEANS B RAISED TO THE POWER 
OF "A" IS = TO N.
SO IN EITHER FORM WE NEED TO BE 
ABLE TO IDENTIFY THE BASE,
WHICH WOULD BE B,
THE EXPONENT, WHICH WOULD BE A,
AND THE NUMBER, 
WHICH WOULD BE N.
I THINK IT'S HELPFUL TO REMEMBER 
A LOGARITHM IS AN EXPONENT.
THAT'S WHY THIS LOGARITHM 
IS EQUAL TO "A,"
WHICH IN EXPONENTIAL FORM 
WOULD BE THIS EXPONENT HERE.
SO WE KNOW EACH OF OUR 
LOGARITHMIC EQUATIONS
WILL HAVE A LOG IN IT.
SO WE'LL GO AHEAD 
AND WRITE DOWN LOG.
AND THEN FROM HERE 
4 WOULD BE THE BASE,
SO THAT'S GOING TO BE 
IN THIS LOWER POSITION HERE.
A LOGARITHM IS AN EXPONENT 
AND "IS" MEANS EQUALS.
SINCE 2 IS THE EXPONENT, 
THIS LOG IS GOING TO EQUAL TO.
AND THEN THE NUMBER WHAT IT'S = 
TO WOULD BE 16, WHICH GOES HERE.
SO 4 TO THE SECOND = 16
IS EQUIVALENT TO 
LOG BASE 4 OF 16 = 2.
NOW, THERE IS ONE MORE THING 
I SHOULD MENTION HERE,
SOMETIMES YOU'LL SEE THIS 
WRITTEN AS 2 = LOG BASE 4 OF 16,
WHICH OF COURSE 
IS THE SAME EQUATION,
IT'S JUST THE LEFT AND RIGHT 
SIDE HAVE BEEN SWITCHED.
WHERE 2 TO THE FIFTH = 32, 
WE NEED A LOGARITHM.
WE KNOW THE BASE IS GOING TO BE 
2, A LOGARITHM IS AN EXPONENT,
THE EXPONENT IS 5, 
SO THE LOG = 5.
AND THEN 32 IS THE NUMBER, 
WHICH WOULD GO HERE,
SO WE HAVE LOG BASE 2 OF 32 
= 35.
SO THE NEXT TWO EXAMPLES, AGAIN, 
WE NEED A LOGARITHM.
THE BASE IN THIS CASE 
WOULD BE 27.
A LOG IS AN EXPONENT, 
OUR EXPONENT IS 1/3,
AND IT'S = TO 3, 
WHICH WOULD BE OUR NUMBER.
SO WE HAVE LOG BASE 27 OF 3 
= 1/3.
AND FOR OUR LAST EXAMPLE,
THE BASE IS 7.
A LOGARITHM IS AN EXPONENT, 
THE EXPONENT IS -2,
SO THE LOG = -2, 
AND IT'S = TO 1/49.
SO WE HAVE 1/49 AS OUR NUMBER.
SO IT DOES TAKE SOME PRACTICE 
TO GET USED TO CONVERTING
BETWEEN LOG FORM 
AND EXPONENTIAL FORM
AND VISE VERSA.
SO I WOULD RECOMMEND 
HAVING THESE TWO EQUATIONS HERE
IN YOUR NOTES.
OKAY, I HOPE YOU FOUND THIS 
HELPFUL.
