- WE WANT TO SOLVE EACH EXPONENTIAL EQUATION
BY WRITING A LOG EQUATION
AND THEN USING THE CHANGE OF BASE FORMULA FOR LOGARITHMS.
I'VE GIVEN THE CHANGE OF BASE FORMULA HERE AT THE BOTTOM
IN RED.
REMEMBER THE CHANGE OF BASE FORMULA GIVES US A WAY
TO EVALUATE ANY LOGARITHM ON THE CALCULATOR
BY TAKING THE LOG OF THE NUMBER
AND DIVIDING BY THE LOG OF THE BASE.
I'VE ONLY GIVEN THE CHANGE OF BASE FORMULA FOR COMMON LOG
AND NATURAL LOG
BECAUSE THESE ARE THE TWO BASES ON THE CALCULATOR
BUT THE CHANGE OF BASE FORMULA IS TRUE FOR ANY BASE.
SO TO SOLVE 5 RAISED TO THE POWER OF X = 476,
WE'RE GOING TO WRITE THIS AS A LOG EQUATION.
SO WE KNOW WE'LL HAVE A LOGARITHM.
THE BASE IS 5, SO THIS WILL BE 5 DOWN HERE.
A LOGARITHM IS AN EXPONENT SO OUR EXPONENT IS X
SO THE LOG IS EQUAL TO X AND OUR NUMBER IS 476.
SO WE HAVE LOG BASE 5 OF 476.
AND NOW, THE DIRECTIONS ARE TELLING US TO EVALUATE THIS LOG
USING THE CHANGE OF BASE FORMULA.
LET'S GO AHEAD AND USE NATURAL LOG
SO THAT MEANS THAT X IS GOING TO BE EQUAL
TO NATURAL LOG OF THE NUMBER 476
DIVIDED BY THE NATURAL LOG OF THE BASE.
SO WE HAVE NATURAL LOG 5.
SO NOW WE'LL GO TO THE CALCULATOR
AND DETERMINE THE APPROXIMATE VALUE OF X.
NATURAL LOG 476 DIVIDED BY NATURAL LOG 5.
SO X IS APPROXIMATELY 3.8308,
WHICH MEANS IF WE TAKE 5 AND RAISE IT TO THIS POWER
IT SHOULD BE APPROXIMATELY EQUAL TO 476
AND LET'S GO AHEAD AND CHECK THAT.
AND WE DID ROUND UP SLIGHTLY,
THEREFORE THE VALUE IS JUST A LITTLE BIT LARGER THAN 476
BUT THIS DOES VERIFY WE SOLVED THIS EQUATION CORRECTLY.
FROM LOOKING AT OUR SECOND EQUATION
OUR FIRST STEP IS GOING TO BE TO ISOLATE
THE EXPONENTIAL PART OF THE EQUATION
SO WE NEED TO DIVIDE BOTH SIDES OF THE EQUATION BY 4.
SO THIS SIMPLIFIES TO 1 SO WE HAVE 3 TO THE POWER OF X
EQUALS THIS QUOTIENT HERE,
984 DIVIDED BY 4 IS EQUAL TO 246.
AND NOW, IT'S JUST LIKE OUR FIRST EXAMPLE,
FROM HERE WE'LL WRITE A LOG EQUATION
SO WE KNOW WE'RE GOING TO HAVE A LOGARITHM.
THE BASE IS 3, A LOGARITHM IS AN EXPONENT
SO THE LOG IS GOING TO EQUAL OUR EXPONENT X
AND OUR NUMBER IS 246
AND NOW WE'LL EVALUATE THIS USING THE CHANGE OF BASE FORMULA
AS THE DIRECTIONS TELL US.
SO X IS GOING TO BE EQUAL TO--
LET'S GO AHEAD AND USE NATURAL LOG AGAIN.
WE'LL HAVE NATURAL LOG OF 246 DIVIDED BY NATURAL LOG OF 3.
SO NATURAL LOG 246 DIVIDED BY NATURAL LOG 3,
X IS APPROXIMATELY 5.0112.
AND GOING BACK TO THE ORIGINAL EQUATION
THIS MEANS 4 x 3 TO THE POWER OF 5.0112
SHOULD BE APPROXIMATELY EQUAL TO 984
AND LET'S GO AHEAD AND CHECK THAT AS WELL.
NOTICE THE VALUE IS VERY CLOSE TO 984,
THEREFORE, THIS DOES VERIFY OUR SOLUTION X
IS APPROXIMATELY 5.0012.
WE'LL ALSO SOLVE EQUATIONS LIKE THIS
USING THE PROPERTIES OF LOGARITHMS
RATHER THAN THE CHANGE OF BASE FORMULA IN ANOTHER VIDEO.
I HOPE YOU FOUND THIS HELPFUL.