- WELCOME TO AN INTRODUCTION 
TO LOGARITHMS.
THE GOALS OF THIS VIDEO 
ARE TO DEFINE A LOGARITHM
AND CONVERT 
EXPONENTIAL EQUATIONS
TO LOGARITHMIC EQUATIONS, 
AND VISE VERSA.
SO THE FIRST THING 
I WANT TO TALK ABOUT IS
WHAT IS A LOGARITHM?
WELL, LOGARITHM 
IS JUST AN EXPONENT.
SO IF SOMEONE ASKS YOU THAT, 
NOW YOU HAVE THE ANSWER.
LET'S TAKE A LOOK 
AT A MORE FORMAL DEFINITION.
LOG BASE A OF X = Y MEANS 
A TO THE POWER OF Y IS = TO X.
SO SINCE A LOGARITHM 
IS AN EXPONENT,
IT SHOULD SEEM LOGICAL THAT
WE CAN WRITE A LOG EQUATION
AS AN EXPONENTIAL EQUATION.
NOTICE "A" WOULD BE THE BASE,
Y WOULD BE THE EXPONENT,
AND X WOULD BE THE NUMBER.
WE CAN SAY THAT Y IS THE POWER, 
WHICH WE RAISE "A" TO GET X.
AND WE SHOULD NOTE THAT "A" 
HAS TO BE GREATER THAN 0
AND "A" CANNOT = 1.
SO IF WE HAVE LOG BASE 2 OF 8 
= 3
THIS MEANS 2 TO THE POWER OF 3 
= 8.
AND A NICE WAY TO REMEMBER THAT 
IS JUST START WITH THE BASE
AND KIND OF 
GO AROUND THE CIRCLE.
2 TO THE POWER OF 3 IS = TO 8.
AND WE CAN USE THIS DEFINITION
IN THE OPPOSITE DIRECTION 
AS WELL.
IF WE HAVE 5 TO THE SECOND = 25
WE CAN REWRITE THIS 
AS A LOG EQUATION
WHERE WE'D HAVE A LOG BASE IS 
= TO 5, THE NUMBER IS 25,
AND THIS MUST = 2 
SINCE 5 TO THE SECOND = 25.
WE'LL COME BACK TO THIS IDEA 
LATER,
BUT LET'S GO AHEAD AND TALK 
ABOUT COMMON LOG
AND NATURAL LOG,
BECAUSE THESE ARE THE TWO LOGS
THAT YOU'LL FIND 
ON YOUR CALCULATOR.
THE LOGARITHM WITH BASE 10 
IS CALLED THE COMMON LOG.
THIS IS THE LOG KEY 
ON YOUR CALCULATOR,
SO YOU WON'T SEE COMMON LOG 
WRITTEN LIKE THIS.
IF IT'S BASE 10 
YOU LEAVE THE BASE OFF.
SO IF YOU HAVE YOUR CALCULATOR 
YOU'LL NOTICE A LOG BUTTON
HERE LEFT SIDE.
SO IF I PRESS LOG 10,000
THIS IS = TO 4.
LET'S GO AHEAD 
AND WRITE THAT DOWN.
AND IF I ASKED YOU 
WHY IS LOG 10,000 = TO 4,
YOU SHOULD BE ABLE TO TELL ME 
THAT, OH, THIS IS COMMON LOG,
WHICH MEANS THAT'S BASE 10.
SO WHAT THIS MEANS IS 10 
TO THE POWER OF 4 MUST = 10,000.
SO IT'S GREAT TO USE 
THE CALCULATOR
TO EVALUATE COMMON LOG,
BUT I THINK IT'S IMPORTANT
THAT YOU UNDERSTAND 
WHAT YOU'RE FINDING.
ANOTHER WORDS, 
YOU'RE FINDING THE EXPONENT
FOR WHAT YOU RAISED 10 TO, 
TO GET THE NUMBER 10,000.
THE SECOND LOG THE CALCULATOR 
HAS IS THE NATURAL LOGARITHM.
THE LOGARITHM WITH BASE E
IS CALLED THE NATURAL 
LOGARITHMIC FUNCTION.
THIS LOG IS ALSO 
ON YOUR CALCULATOR
AS CAPITAL "L", CAPITAL "N".
SO YOU WON'T SEE LOG BASE 
E OF X, ITS WRITTEN LNX.
SO YOU HAVE TO REMEMBER THAT 
THIS JUST MEANS LOG BASE E.
REMEMBER E IS A IRRATIONAL 
NUMBER SIMILAR TO PI
EXCEPT ITS APPROXIMATELY 
EQUAL TO 2.718.
SO FOR EXAMPLE, 
IF WE TYPE IN NATURAL LOG OF 95
ON THE CALCULATOR,
HERE'S THE NATURAL LOG KEY.
WE'RE GOING TO GET A DECIMAL 
APPROXIMATION.
SO IT'S APPROXIMATELY 4.5539.
SO WHAT DOES THAT MEAN?
REMEMBER NATURAL LOG MEANS 
LOG BASE E,
SO THAT MEANS E TO THE POWER OF 
4.5539 IS APPROXIMATELY = TO 95.
AND WE DID ROUND THIS, 
SO IT WON'T BE EXACT.
BUT LET'S GO AHEAD AND CHECK IT.
SO IF WE PRESS SECOND NATURAL 
LOG, IT BRINGS UP BASE E.
WE CAN JUST TYPE 
IN OUR EXPONENT,
AND YOU CAN SEE 
IT'S APPROXIMATELY 95.
LET'S GO AND TAKE A LOOK 
AT A FEW MORE EXAMPLES.
LET'S TAKE THESE 
LOGARITHMIC EQUATIONS
AND WRITE THEM 
AS EXPONENTIAL EQUATIONS,
WHICH MEANS WE'RE JUST GOING 
TO REWRITE THESE
USING THE DEFINITION 
OF A LOGARITHM.
SO HERE'S A DEFINITION,
BUT IF YOU CAN JUST REMEMBER TO 
GO CLOCKWISE AROUND THE EQUATION
2 TO THE THIRD MUST = 8.
THE NICE THING ABOUT GOING FROM 
LOG FORM TO EXPONENTIAL FORM,
YOU CAN CHECK IT.
THIS IS TRUE, 
THEREFORE IT'S CORRECT.
HERE WE'D HAVE 
3 TO THE POWER OF -2 = 1/9.
THAT'S CORRECT.
HERE'S NATURAL LOG, 
SO THAT WOULD BASE E.
IT MIGHT BE HELPFUL TO WRITE 
THAT IN.
E TO THE FIRST POWER MUST = E.
AND THAT'S CORRECT.
AND THEN ANOTHER COMMON LOG 
RECOGNIZING THIS AS BASE 10,
10 TO THE POWER OF 3 
MUST = 1,000.
AND LET'S GO AHEAD 
AND TAKE A LOOK AT A COUPLE MORE
IN THE OPPOSITE DIRECTION.
WE WANT TO WRITE THESE 
EXPONENTIAL EQUATIONS
AS LOGARITHMIC EQUATIONS,
SO ON THIS FIRST PROBLEM WE KNOW 
WE'RE GOING TO HAVE A LOGARITHM.
SO THERE'S THREE KEY COMPONENTS, 
THE BASE, THE EXPONENT,
AND THE NUMBER.
WELL, THE BASE IS 4 
DOWN THERE IN THE BASEMENT.
YOU CAN THINK OF IT BEING 
IN A LOWER POSITION.
A LOGARITHM IS AN EXPONENT,
SO THE EXPONENT IS 3, 
THE LOG MUST = 3,
AND THE NUMBER IS 64.
AGAIN, 4 TO THE POWER OF 3 = 64, 
AND IT CHECKS.
THE THIRD ONE, AGAIN, 
WE'RE GOING TO HAVE A LOGARITHM.
HERE IT'S BASE 10.
A LOGARITHM IS AN EXPONENT, 
SO THE EXPONENT IS -2,
THE LOG IS = TO -2,
AND THEN THE NUMBER IS 100.
NOW, THERE'S ONE MORE THING
ON THIS ONE THOUGH,
IT IS A COMMON LOG.
SO IT'S NOT WRONG 
TO PUT THIS BASE 10 IN THERE,
BUT YOU'LL VERY SELDOM SEE IT,
SO I'M GOING TO GO AHEAD 
AND ERASE THAT.
REMEMBER IF THE BASE ISN'T THERE 
WE KNOW ITS COMMON LOG.
AND THE LAST ONE, 
IT'S A LOGARITHM,
IT'LL HAVE A LOG.
NOW IT'S BASE E.
AGAIN, A LOGARITHM 
IS AN EXPONENT, SO IT MUST = 2,
AND THE NUMBER IS APPROXIMATELY 
7.389.
NOW, IN THIS ONE THOUGH,
WE HAVE LOG BASE E.
WE CAN'T LEAVE IT IN THIS FORM.
WE SHOULD REWRITE THIS 
AS NATURAL LOG 7.389.
SO THIS WOULD BE 
OUR INTERMEDIATE STEP,
BUT WE WANT TO LEAVE THIS 
AS OUR FINAL ANSWER.
OKAY, NOW YOU MIGHT BE ASKING
WHAT'S THE BIG DEAL 
WITH LOGARITHMS.
WELL, THERE ARE A LOT OF 
LOGARITHMIC SCALES
USED IN REAL LIFE.
THREE OF THE MOST COMMON ONES 
ARE THE RICHTER SCALE
TO MEASURE THE INTENSITY 
OF EARTHQUAKES
IS A LOGARITHMIC SCALE.
A DECIBLE SCALE TO MEASURE SOUND 
LEVELS IS A LOGARITHMIC SCALE,
AS WELL AS THE PH SCALE
TO MEASURE HOW BASIC OR ACIDIC 
A SUBSTANCE IS.
THAT IS ALSO 
A LOGARITHMIC SCALE.
SO IF YOU SEARCH ANY OF THESE 
ONLINE,
YOU'LL NOTICE THAT 
THEY ALL INVOLVE LOGARITHMS.
OKAY, SO THIS IS THE FIRST 
OF SEVERAL VIDEOS ON LOGARITHMS.
I HOPE YOU FOUND IT HELPFUL.
THANK YOU FOR WATCHING.
