- WE WANT TO SOLVE THE GIVEN
EXPONENTIAL EQUATIONS
USING LOG RHYTHMS.
IN GENERAL, TO SOLVE
AN EXPONENTIAL EQUATION
WITH ONE EXPONENTIAL PART,
YOU WANT TO ISOLATE
THE EXPONENTIAL PART
AND THEN TAKE IT TO
THE COMMON LOG OR NATURAL LOG
ON BOTH SIDES OF THE EQUATION
AND THIS WILL ALLOW US
TO SOLVE FOR X.
LOOKING A OUR FIRST EXAMPLE,
WE HAVE 7 RAISED TO THE POWER
OF 4X PLUS 1 EQUALS 128.
NOTICE HOW THE EXPONENTIAL
PART IS ALREADY ISOLATED
ON THE LEFT SIDE
OF THE EQUATION.
SO, WE CAN START
BY TAKING THE LOG
OF BOTH SIDES OF THE EQUATION.
I'M GOING TO TAKE THE NATURAL
LOG OF BOTH SIDES.
SO, WE'LL HAVE THE NATURAL LOG
OF 7 RAISED TO THE POWER OF 4X
PLUS 1 EQUALS THE NATURAL LOG
OF 128.
AND NOW SINCE OUR GOAL
IS TO SOLVE FOR X,
WE CAN MOVE X
OUT OF THE EXPONENT POSITION
BY APPLYING THE POWER PROPERTY
OF LOG RHYTHMS,
WHICH MEANS WE CAN TAKE
THIS EXPONENT HERE
AND MOVE IT TO THE FRONT.
SO, IT'S GOING TO BE
THE QUANTITY 4X PLUS 1
TIMES NATURAL LOG 7.
AND NOW, WE WANT TO SOLVE
FOR X.
SO, WHAT WE'RE GOING TO DO
HERE IS CLEAR THE PARENTHESES
BY DISTRIBUTING NATURAL LOG 7
AND THEN SOLVE FOR X.
SO, WE'LL HAVE 4X
NATURAL LOG 7,
PLUS NATURAL LOG 7
EQUALS NATURAL LOG 128.
NOW, THIS MAY LOOK
A LITTLE STRANGE,
BUT THIS IS REALLY JUST
A BASIC TWO-STEP EQUATION.
TO SOLVE THIS EQUATION FOR X,
WE'LL START BY UNDOING
THIS ADDITION
BY SUBTRACTING NATURAL LOG 7
ON BOTH SIDES OF THE EQUATION.
THIS WOULD BE ZERO,
SO WE HAVE 4X NATURAL LOG 7,
EQUALS NATURAL LOG 128
MINUS NATURAL LOG 7.
REMEMBER, WE CANNOT
SUBTRACT THESE
BECAUSE THESE ARE NOT LIKE
LOG RHYTHMS.
AND NOW TO SOLVE FOR X,
WE NEED TO DIVIDE.
SO, WE'LL DIVIDE BY 4
NATURAL 7 ON BOTH SIDES.
NOTICE ON THE LEFT SIDE,
4 OVER 4 SIMPLIFIES TO 1,
NATURAL 7 OVER NATURAL 7
SIMPLIFIES TO 1.
SO, WE HAVE X EQUALS
THIS QUOTIENT,
BUT NOW WE'LL HAVE TO USE
THE CALCULATOR
TO GET A DECIMAL
APPROXIMATION.
SO, WE'LL HAVE A SET
OF PARENTHESIS
AROUND THE NUMERATOR
AND THE DENOMINATOR.
SO, WE'LL HAVE
AN OPEN PARENTHESIS,
NATURAL LOG 128
MINUS NATURAL LOG 7.
SO, THERE'S THE NUMERATOR
DIVIDED BY OUR DENOMINATOR
OF 4 NATURAL LOG 7.
AND IT'S TYPICAL TO ROUND
TO FOUR DECIMAL PLACES.
SO, X IS APPROXIMATELY 0.3734.
AND TO CHECK THIS, WE WITH
TO RAISE 7 TO THE POWER OF 4
TIMES 0.3734 PLUS 1
AND THIS SHOULD BE
APPROXIMATELY EQUAL TO 129.
LET'S GO AHEAD AND CHECK
TO MAKE SURE
THAT OUR SOLUTION HERE
IS CORRECT.
WE DID ROUND UP SLIGHTLY,
THEREFORE, THE VALUE IS
A LITTLE BIT LARGER THAN 128,
BUT THIS DOES VERIFY
OUR SOLUTION.
LET'S TAKE A LOOK
AT ANOTHER EXAMPLE.
HERE, WE HAVE E RAISED TO THE
POWER OF 2X MINUS 5 EQUALS 61.
AGAIN, THE EXPONENTIAL PART
IS ALREADY ISOLATED.
SINCE HERE WE HAVE BASE E,
IT'S GOING TO BE EASIER
IF WE USE THE NATURAL LOG
RATHER THAN THE COMMON LOG.
SO, IF WE TAKE THE NATURAL LOG
OF BOTH SIDES OF THE EQUATION,
WE WOULD HAVE NATURAL LOG OF E
RAISED TO THE POWER OF 2X
MINUS 5 EQUALS NATURAL LOG 61.
AND AGAIN, WE CAN MOVE X
OUT OF THE POSITION
OF THE EXPONENT
BY APPLYING THE POWER PROPERTY
OF LOG RHYTHMS.
SO, WE'LL MOVE THIS
TO THE FRONT.
SO, NOW WE HAVE THE QUANTITY
2X MINUS 5 TIMES NATURAL LOG E
EQUALS NATURAL LOG 61.
NOW, NOTICE
IN THE PREVIOUS EXAMPLE,
WE DISTRIBUTED THE NATURAL LOG
AND THEN, SOLVE FOR X.
BUT NATURAL LOG E
IS EQUAL TO 1.
REMEMBER NATURAL LOG E
IS LOG BASE E.
SO, THIS SIMPLIFIES
TO THE EXPONENT,
WHICH WE WOULD RAISE
THIS BASE E TO,
TO OBTAIN E
WHICH WOULD BE EQUAL TO 1.
E TO THE FIRST IS EQUAL TO
IF YOU'RE STILL NOT CONVINCED,
OF COURSE WE COULD DO THIS
ON THE CALCULATOR.
NATURAL LOG E IS EQUAL TO 1
AND SINCE THIS SIMPLIFIES
TO 1,
WE JUST HAVE THE QUANTITY 2X
MINUS 5 EQUALS NATURAL LOG 61,
SO WE'LL ADD 5 TO BOTH SIDES
OF THE EQUATION.
THIS WOULD BE 2X EQUALS,
THIS WOULD BE ZERO.
NATURAL LOG 61 PLUS 5
AND THEN DIVIDE BOTH SIDES BY
SO, WE HAVE X EQUALS
THIS QUOTIENT HERE.
WE'LL GO AHEAD AND GET
A DECIMAL APPROXIMATION.
NATURAL LOG 61 PLUS 5,
DIVIDE ALL THIS BY 2.
SO, X IS APPROXIMATELY 4.5554.
WHICH MEANS, IF WE TAKE E
AND RAISE IT TO THE POWER OF 2
TIMES 4.5554 AND SUBTRACT 5,
IT SHOULD BE APPROXIMATELY
EQUAL TO 61.
LET'S GO AHEAD AND CHECK THAT.
SO, WE PRESS
SECOND NATURAL LOG,
BRINGS UP E
RAISED THE POWER OF
AND THEN WE'LL HAVE 2
TIMES 4.5554 MINUS 5.
ALL THIS IS OUR EXPONENT
AND THIS IS APPROXIMATELY
EQUAL TO 61.
OKAY. THAT'S GOING TO DO IT
FOR THESE TWO EXAMPLES.
WE'LL TAKE A LOOK AT TWO MORE
EXAMPLES IN THE NEXT VIDEO.
