- 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.