- WE WANT TO SOLVE THE EXPONENTIAL EQUATION.
WE WANT TO GIVE THE EXACT SOLUTION
AND THE SOLUTION ROUNDED TO FOUR DECIMAL PLACES.
SO OUR STRATEGY WILL BE TO ISOLATE THE EXPONENTIAL PART
AND THEN TAKE THE LOG OF BOTH SIDES OF THE EQUATION.
TO ISOLATE THE EXPONENTIAL PART,
WE WANT TO ISOLATE 1.39 RAISE TO THE POWER OF T.
NOTICE TO DO THIS WE WILL SUBTRACT 17 ON BOTH SIDES
AND THEN DIVIDE BY -100.
SO OUR FIRST STEP WILL BE TO SUBTRACT 17 FROM BOTH SIDES.
THIS WOULD BE 0, SO WE HAVE -100 x 1.39
RAISE TO THE POWER OF T = -3,083 - 17 WOULD BE -3,100
AND THEN NOW WE'LL DIVIDE BOTH SIDES BY A -100.
SO NOTICE THIS SIMPLIFIES TO 1.
SO WE HAVE 1.39 RAISE TO THE POWER OF T = -3,100
DIVIDED BY -100 WOULD BE 31
AND NOW IF WE TAKE THE LOG OF BOTH SIDES OF THE EQUATION,
WE CAN APPLY THE POWER PROPERTY OF LOG RHYTHMS,
AND THEN SOLVE FOR T.
WE WILL END UP USING A CALCULATOR
TO GET A DECIMAL APPROXIMATION.
SO WE HAVE THE OPTION OF USING
EITHER A NATURAL LOG OR A COMMON LOG.
FOR THIS EXAMPLE,
LET'S TAKE THE COMMON LOG OF BOTH SIDES.
NOW WE'LL APPLY THE POWER OF PROPERTY
OF LOG RHYTHMS TO THE LEFT
MEANING WE CAN TALK THE EXPONENT OF T AND MOVE IT TO THE FRONT
GIVING US T x LOG 1.39 = LOG 31.
NOW TO SOLVE FOR T,
WE'LL DIVIDE BOTH SIDES BY A LOG 1.39.
NOTICE ON THE LEFT, THIS SIMPLIFIES TO 1.
SO WE HAVE T = THIS QUOTIENT HERE
WHERE WE HAVE LOG 31 DIVIDED BY LOG 1.39.
SO THIS WOULD BE OUR EXACT SOLUTION
AND REMEMBER BOTH OF THESE ARE COMMON LOG
MEANING THEY'RE LOG-BASE 10 SO NOW WE CAN GO TO OUR CALCULATOR
TO GET OUT DECIMAL APPROXIMATION OR THE SOLUTION
ROUNDED TO FOUR DECIMAL PLACES.
WE HAVE LOG 31 DIVIDED BY LOG 1.39.
CONVERTING TO FOUR DECIMAL PLACES
WE WOULD HAVE T AS APPROXIMATELY 10.4280.
LET'S TAKE A MOMENT AND CHECK OUR SOLUTION
BY SUBBING THIS APPROXIMATE VALUE FOR T
BACK IN TO THE ORIGINAL EQUATION.
MEANING 17 - 100 x 1.39 RAISE TO THE POWER OF 10.4280
SHOULD GIVE US APPROXIMATELY -3,083.
NOTICE HOW THIS VALUE IS OFF SLIGHTLY
BUT IT IS VERY CLOSE TO -3,083 AND IT'S OFF SLIGHTLY
BECAUSE WE DID USE THE SOLUTION ROUNDED TO FOUR DECIMAL PLACES.
I HOPE YOU FOUND THIS HELPFUL.