- SOLVING EQUATIONS.
THESE WILL HAVE
THE SAME BASES.
WE ARE GOING TO DEAL
WITH A RULE THAT SAYS
IF WE TAKE THE LOG, BASE "A,"
OF "A" TO THE X POWER.
AND IT'S THE SAME "A,"
THEN EVERYTHING CANCELS OUT
AND WE END UP WITH JUST X.
THIS IS BECAUSE OF LOG
IN AN EXPONENT FUNCTION
ARE INVERSES OF ONE ANOTHER.
THIS IS ONE OF THE EASIER
RULES, VERY NICELY.
SO WE'LL START WITH A NICE
SIMPLE BEGINNING PROBLEM,
LOG BASE 12 OF 12 TO THE 5th
POWER IS JUST EQUAL TO 5.
THE LOG BASE 12 INTO 12 SORT
OF UNRAVEL ONE ANOTHER
AND WE JUST END UP WITH FIVE.
AGAIN THIS IS BECAUSE THEY'RE
INVERSES OF ONE ANOTHER.
THIS NEXT EXAMPLE,
THE LOG BASE 2 OF 128.
NOW WE CAN SEE THAT THE BASE 2
HERE IS NOT THE SAME AS 128.
HOWEVER, IF I PRIME FACTORIZE
128,
MEANING I BREAK IT DOWN
WITH ITS TREE, 128
AND I DIVIDE IT BY 2.
64 TIMES 2 AND I CAN BREAK
64 DOWN TO 2 TIMES 32,
I'M GOING TO END UP
WITH A WHOLE BUNCH OF TWOS.
IN FACT, IF YOU KEEP BREAKING
THAT DOWN WE WILL END UP
WITH LOG BASE 2
OF 2 RAISED TO THE 7th POWER.
16 HERE ACTUALLY BREAKS DOWN
TO FOUR MORE TWOS.
SO WE'LL HAVE THE THREE TWOS
ALONG THE LINE HERE.
AND THEN WE'LL HAVE THE 16.
AND VERY NICELY WE HAVE
THE SAME BASE NOW.
WE HAVE LOG BASE 2 OF 2
TO THE 7th POWER
AND EVERYTHING COLLAPSES DOWN
TO JUST 7.
THIS IS REALLY THE BEST WAY
TO GO
IF YOU CAN MAKE THE BASES
MATCH LIKE THAT.
OFTEN--MORE OFTEN
THEN NOT YOU CANNOT.
AND WE'LL JUST TRY A COUPLE
MORE EXAMPLES.
LOG BASE THREE--1 OVER 81.
NOW, 1 OVER 81 OBVIOUSLY
IS NOT 3.
HOWEVER, 81 BREAKS DOWN
AS WELL.
WE CAN DO A LITTLE TREE OFF
TO THE SIDE IF WE NEEDED TO.
AND WE WOULD SEE THAT 81
IS THE SAME THING
AS 3 RAISED TO THE 4th POWER.
NOW, ONE OVER 3
TO THE 4th POWER
IS STILL NOT BASE 3 QUITE YET.
WE'D NEED TO DO
A LITTLE MORE MANIPULATION--
WE'LL HAVE BASE 3.
AND IF YOU THINK BACK
TO YOUR EXPONENT RULES,
IF YOU HAVE A POSITIVE
EXPONENT IN THE DENOMINATOR
THAT MEANS YOU CAN ACTUALLY
BRING IT UP TO THE NUMERATOR
AND IT BECOMES
A NEGATIVE EXPONENT.
SO THIS IS, LIKE, THREE
TO THE NEGATIVE 4th OVER ONE.
IT'S LIKE I TOOK
THE RECIPROCAL.
AND VERY NICELY WE CAN SEE
LOG BASE THREE OF THREE
TO THE NEGATIVE 4th POWER,
EVERYTHING COLLAPSES DOWN
VERY NICELY
AND YOU GET NEGATIVE FOUR.
SOME OF THESE YOU WILL HAVE TO
DO THE TREE
LIKE WE DID TO THE 128
UP ABOVE
TO SEE IF IT REALLY
IS THE RIGHT BASE OR NOT.
IF IT'S NOT THEN THIS RULE
THAT WE ARE USING RIGHT NOW
WILL NOT APPLY.
LET'S DO ONE MORE EXAMPLE,
THIS WILL BE A FANCY ONE.
LOG BASE NINE OF THREE.
NOW, YOU CAN KIND OF SEE
THAT THERE'S A CONNECTION
BETWEEN THE THREE AND THE NINE
AND THERE IS.
AND WE GOT TO GET
A LITTLE TRICKY HERE.
AND I CAN'T MESS
WITH THE BASE.
SO I CAN'T CHANGE
THE NINE DOWN HERE,
THAT'S PART OF THE PROBLEM.
HOWEVER, I CAN DO
SOME MANIPULATION
TO THE THREE ON THE INSIDE
AND THE CONNECTION HERE--
LOG BASE NINE OF--
THREE IS THE SAME THING
AS THE SQUARE ROOT OF NINE.
NOW, YOU'VE LEARNED
THIS SEMESTER SOME NEW RULES
WITH EXPONENTS.
AND WHEN YOU HAVE
THE SQUARE ROOT OF NINE,
THAT'S THE SAME THING AS
SAYING THAT THIS IS NOW NINE
RAISED TO THE ONE HALF POWER.
AND NOW WE HAVE, FINALLY,
LOG BASE NINE OF NINE
RAISED TO THE ONE HALF POWER.
SO THE NINES, AGAIN,
UNRAVEL ONE ANOTHER.
WE END UP WITH A ONE HALF.
AND THERE YOU HAVE IT.
