- WE WANT TO SOLVE THE GIVEN
EXPONENTIAL EQUATIONS
USING THE PROPERTIES
OF EXPONENTS.
THEY'RE ASKING US TO SOLVE FOR X
USING LIKE BASES,
WHICH MEANS YOU WON'T
HAVE TO USE LOGARITHMS
TO SOLVE THESE EQUATIONS.
IN GENERAL,
IF A TO THE POWER OF M
IS EQUAL TO A TO THE POWER OF N,
THEN M IS EQUAL TO N.
WHICH MEANS IF THESE TWO
WERE EQUAL TO EACH OTHER
AND THE BASES ARE THE SAME,
THEN THE EXPONENTS MUST BE EQUAL
TO EACH OTHER.
SO LOOKING AT OUR
FIRST EQUATION,
WE HAVE 2 RAISED TO THE POWER OF
X - 3
= 2 RAISED TO THE POWER
OF 5X + 11.
SINCE OUR BASES ARE THE SAME
AND THESE ARE EQUAL
TO EACH OTHER,
X - 3 MUST = 5X + 11.
AND NOW WE JUST NEED TO SOLVE
THIS EQUATION FOR X
AND WE'LL HAVE OUR SOLUTION.
WE NEED TO GET X
ON ONE SIDE OF THE EQUATION,
SO WE NEED TO EITHER SUBTRACT X
ON BOTH SIDES,
OR SUBTRACT 5X ON BOTH SIDES.
LET'S GO AHEAD AND SUBTRACT X
ON BOTH SIDES OF THE EQUATION,
SO IT GIVES US -3 = 4X + 11.
AND WE'LL SUBTRACT 11
ON BOTH SIDES.
IF -14 = THIS WOULD BE 4X,
THIS IS 0,
DIVIDE BOTH SIDES BY 4.
THIS SIMPLIFIES,
SO WE HAVE -14/4 = X.
THIS SIMPLIFIES, THEY BOTH
CONTAIN A COMMON FACTOR OF 2.
SO WE HAVE -7/2 = X,
OR, IF WE WANT,
WE CAN WRITE THIS AS X = -7/2.
THIS SECOND EXAMPLE
IS A LITTLE BIT MORE CHALLENGING
BECAUSE OF THIS FRACTION.
REMEMBER WE CAN VIEW THIS
AS 1/3 TO THE POWER OF 1.
AND TO WRITE THIS SO THAT
BOTH SIDES OF THE EQUATION
HAVE A BASE OF 3,
WE CAN MOVE THIS UP
INTO THE NUMERATOR.
REMEMBER THAT IS GOING TO CHANGE
THE SIGN OF THE EXPONENT,
SO NOW WE WOULD HAVE 3
RAISED TO THE POWER OF -1
ALL RAISED TO THE POWER X + 2 =
3 RAISED TO THE POWER OF 4X + 6.
NOW, HERE WE HAVE A POWER
RAISED TO A POWER,
WHICH MEANS WE CAN MULTIPLY
THESE EXPONENTS.
SO WE CAN WRITE THIS AS 3
RAISED TO THE POWER OF -1
x THE QUANTITY X + 2
= 3 RAISED TO THE POWER OF
4X + 6.
SO FINALLY NOW WE HAVE
THE LEFT SIDE AND RIGHT SIDE
WITH A COMMON BASE, WHICH IS 3.
AND SINCE THEY'RE EQUAL, THE
EXPONENTS MUST ALSO BE EQUAL.
SO -1 x THE QUANTITY X + 2
MUST = 4X + 6.
LET'S GO AHEAD AND CLEAR
OUR PARENTHESIS.
SO WE'LL DISTRIBUTE HERE.
SO WE'LL HAVE -X - 2 = 4X + 6.
LET'S GO AHEAD AND ADD X
TO BOTH SIDES OF THE EQUATION.
AT THE SAME TIME,
LET'S GO AHEAD AND SUBTRACT 6.
THIS SHOULD BE 0,
THIS SHOULD BE -8,
THIS WOULD BE 4X + X IS 5X,
AND THIS IS ALSO 0.
SO NOW WE'LL DIVIDE BOTH SIDES
BY 5, SO WE HAVE -8/5 = X,
OR, AGAIN, IF WE WANT,
WE CAN SWITCH THE ORDER HERE,
AND WRITE X = -8/5.
OKAY, WE'LL LOOK AT SOME MORE
EXAMPLES IN THE NEXT VIDEO.
