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- 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.