- WE WANT TO EVALUATE EACH
OF THE EXPONENTIAL EXPRESSIONS
TO ILLUSTRATE ONE OF
THE PROPERTIES OF LOG RHYTHMS.
SO TO EVALUATE E RAISE TO
THE POWER OF NATURAL LOG 2,
WE'RE GOING TO CREATE
AN EQUATION
BY SETTING THIS EQUAL TO X.
SO IF WE CAN DETERMINE
THE VALUE OF X
WE'LL BE ABLE TO EVALUATE
THIS EXPRESSION.
AND SINCE NOW HAVE
AN EXPONENTIAL EQUATION
WE'LL WRITE THIS AS A LOG
EQUATION TO SOLVE FOR X.
AND IF YOU NEED THE REVIEW,
HERE ARE SOME NOTES BELOW
IN RED
ON HOW TO CONVERT AN EXPONENTIAL
EQUATION TO A LOG EQUATION.
THE MAIN THING IS WE WANT TO
IDENTIFY THE BASE,
THE EXPONENT, AND THE NUMBER.
WE KNOW WE'LL HAVE A LOG IN
OUR LOG EQUATION.
THE BASE IS E,
SO WE'LL HAVE LOG BASE E,
WHICH WE'LL WRITE AS NATURAL LOG
IN JUST A MOMENT.
A LOG RHYTHM IS AN EXPONENT.
HERE OUR EXPONENT
IS NATURAL LOG 2.
AND THIS IS EQUAL TO X SO THE
NUMBER OF PART OF THE LOG IS X.
SO AGAIN, LOG BASE E OF X
IS THE SAME AS NATURAL LOG X.
SO WE HAVE NATURAL LOG X
= NATURAL LOG 2.
SO THESE TWO LOGS ARE EQUAL
TO EACH OTHER
AND THEIR BASES ARE THE SAME.
THEREFORE, THE NUMBER PART
OF THE LOGS MUST BE THE SAME.
SO X = 2.
SO IF X = 2, THEN
THIS EXPRESSION = 2 AS WELL.
LET'S TAKE A LOOK
AT ANOTHER EXAMPLE.
HERE WE HAVE 10 RAISE TO
THE POWER OF COMMON LOG 3.
SO WE'LL SET THIS = TO X
AND WRITE IT AS A LOG EQUATION.
SO WE KNOW WE'LL HAVE A LOG.
IT'S GOING TO BE LOG BASE 10.
A LOG IS AN EXPONENT.
OUR EXPONENT IS COMMON LOG 3.
IT'S EQUAL TO X
SO THE NUMBER PART IS X.
AGAIN, THE RIGHT SIDE, LOG 3
WOULD BE LOG BASE 10 OF 3.
SO AGAIN, THESE ARE EQUAL TO
EACH OTHER,
THEREFORE X = 3,
WHICH MEANS OUR ORIGINAL
EXPRESSION SIMPLIFIES TO 3.
LET'S LOOK AT ONE MORE EXAMPLE
AND THEN WE'LL COME BACK
AND STATE THE SHORTCUT.
WE HAVE 5 RAISE TO THE POWER
OF LOG BASE 5 OF 7.
SO WE'LL SET THIS EQUAL TO X
AND WRITE OUR LOG EQUATION.
OUR LOG WILL HAVE BASE 5.
A LOG IS AN EXPONENT SO THE
EXPONENT IS LOG BASE 5 OF 7.
AND THIS IS EQUAL TO X SO
THE NUMBER PART OF THE LOG IS X.
THEREFORE, X = 7.
NOTICE IN EACH OF THESE 3 CASES
THESE EXPRESSIONS SIMPLIFY
TO JUST THE NUMBER PART
OF THE LOG IN THE EXPONENT.
5 RAISE TO THE POWER OF LOG BASE
5 OR 7 SIMPLIFIED TO 7.
10 RAISE TO THE POWER
OF COMMON LOG 3 SIMPLIFIED TO 3.
AND E NATURAL LOG 2
SIMPLIFIED TO 2.
SO IN GENERAL,
IF WE HAVE BASE B RAISE TO
THE POWER OF LOG BASE B OF X
THIS WILL ALWAYS SIMPLIFY
TO THE NUMBER PART OF THE LOG
OR IN THIS CASE JUST X.
I HOPE THESE EXAMPLES
HELPED EXPLAIN
WHY THIS PROPERTY IS TRUE
AND ONCE WE UNDERSTAND THAT
WE CAN TAKE ADVANTAGE OF IT
TO SIMPLIFY EXPRESSIONS
LIKE THIS.
