- WE WANT TO SOLVE THE GIVEN
EXPONENTIAL EQUATIONS.
TO DO THIS,
WE'LL BE USING LOGARITHMS.
SO IN GENERAL, WHEN SOLVING
AN EXPONENTIAL EQUATION
WITH ONE EXPONENTIAL PART
LIKE THESE EXAMPLES,
WE WANT TO ISOLATE
THE EXPONENTIAL PART,
TAKING THE COMMON LOG
OR NATURAL LOG
OF BOTH SIDES OF THE EQUATION
IN ORDER TO SOLVE FOR X.
SO LOOKING
AT OUR FIRST EXAMPLE
WE HAVE 3 RAISED TO THE POWER
OF X DIVIDED BY 4 = 10.
NOTICE THE EXPONENTIAL PART
IS ALREADY ISOLATED
IN THE LEFT SIDE
OF THE EQUATION.
SO LET'S GO AHEAD
AND TAKE THE NATURAL LOG
OF BOTH SIDES OF THE EQUATION.
NOW, THE REASON WE WANT TO USE
EITHER NATURAL LOG
OR COMMON LOG
IS WE WILL HAVE TO USE
THE CALCULATOR
IN ORDER TO SOLVE
THESE PROBLEMS.
NOW THAT WE'VE TAKEN
THE NATURAL LOG
ON BOTH SIDES OF THE EQUATION,
WE CAN NOW APPLY THE POWER
PROPERTY OF LOGARITHMS
TO THE LEFT SIDE TO MOVE
THE EXPONENT INTO THE FRONT
SO IT BECOMES THE COEFFICIENT.
SO NOW WE CAN WRITE THIS
AS X DIVIDED BY 4
x NATURAL LOG 3
= NATURAL LOG 10.
AND NOW WE'LL SOLVE
THIS EQUATION FOR X.
LETS START BY DIVIDING
BOTH SIDES BY NATURAL LOG 3.
THIS SIMPLIFIES TO 1,
SO WE HAVE X DIVIDED BY 4
= NATURAL LOG 10
DIVIDED BY NATURAL LOG 3.
AND NOW TO SOLVE FOR X,
WE'LL MULTIPLY BOTH SIDES
OF THE EQUATION BY 4,
AND WE CAN THINK OF THIS
AS 4/1.
SO WHEN MULTIPLYING
BY THE RIGHT SIDE,
WE ONLY HAVE TO MULTIPLY
THE NUMERATOR BY 4.
THIS SIMPLIFIES TO 1, SO WE
HAVE X = THIS QUOTIENT HERE,
WHICH WE WILL HAVE TO ROUND.
SO NOW WE'LL GO
TO THE CALCULATOR.
OUR NUMERATOR IS 4
x NATURAL LOG 10
DIVIDED BY NATURAL LOG 3.
SO X IS APPROXIMATELY 8.3836.
AND WE SHOULD CHECK
OUR SOLUTION.
THIS MEANS 3 RAISED TO THE
POWER OF 8.3836 DIVIDED BY 4
SHOULD BE APPROXIMATELY = TO
10 SINCE THIS IS ROUNDED.
3 RAISED TO THE POWER
OF 8.3836 DIVIDED BY 4,
NOTICE HOW IT IS APPROXIMATELY
10, WE DID ROUND DOWN
AND THAT'S THE REASON
WHY IT'S JUST LESS THAN 10.
LET'S TAKE A LOOK
AT OUR SECOND EXAMPLE.
AGAIN, NOTICE THIS IS ONLY
ONE EXPONENTIAL PART
AND ITS ALREADY ISOLATED,
SO WE'LL TAKE EITHER
THE COMMON LOG OR NATURAL LOG
OF BOTH SIDES OF THE EQUATION.
LET'S GO AHEAD AND TAKE
THE NATURAL LOG AGAIN.
SO THE NATURAL LOG OF 2
RAISED TO THE POWER OF -X
DIVIDED BY 5
= THE NATURAL LOG OF 61.
AND NOW WE'LL APPLY THE POWER
PROPERTY OF LOGARITHMS
ON THE LEFT SIDE.
SO WE CAN TAKE THIS EXPONENT
HERE AND MOVE IT TO THE FRONT,
SO NOW WE HAVE -X DIVIDED BY 5
x NATURAL LOG 2
= NATURAL LOG 61.
AND NOT TO SOLVE THIS EQUATION
FOR X,
WE'LL START BY DIVIDING
BOTH SIDES BY NATURAL LOG 2.
THIS SIMPLIFIES TO 1,
SO NOW WE HAVE -X DIVIDED BY 5
= NATURAL LOG 61
DIVIDED BY NATURAL LOG 2.
AND NOW TO UNDO THIS DIVISION
WE'LL MULTIPLY BOTH SIDES
BY 5,
BUT SINCE WE WANT A POSITIVE X
LET'S MULTIPLY BOTH SIDES
BY -5 OR -5/1 IF WE WANT.
NOTICE ON THE LEFT SIDE 5/5
SIMPLIFIES TO 1, -X IS = TO X.
THIS IS GOING TO = -5
NATURAL LOG 61
DIVIDED BY NATURAL LOG 2.
LET'S GO AHEAD AND GET
A DECIMAL APPROXIMATION
FOR THIS.
SO X IS APPROXIMATELY
-29.6537.
AND TO CHECK
THIS WE'LL SUB THIS X VALUE
BACK INTO THE ORIGINAL
EQUATION.
NOTICE HOW WE HAVE
A NEGATIVE IN THE EQUATION
AND X IS NEGATIVE.
SO THIS IS GOING TO BE 2
TO THE POWER OF +29.6537
DIVIDED BY 5 SHOULD BE
APPROXIMATELY = TO 61.
AND AS YOU CAN SEE,
OUR SOLUTION CHECKS.
OKAY, WE'LL TAKE A LOOK
AT SOME MORE EXAMPLES
IN THE NEXT VIDEO.
