- WE WANT TO SOLVE
THE GIVEN EXPONENTIAL EQUATIONS
BY USING LIKE BASES,
WHICH MEANS, YOU WANT TO ISOLATE
THE EXPONENTIAL PART
ON ONE SIDE OF THE EQUATION
AND THEN, WRITE BOTH SIDES OF
THE EQUATION WITH THE SAME BASE.
IF WE CAN DO THIS, THEN
THE EXPONENTS WILL BE EQUAL.
IN GENERAL,
WE SAY IF A TO THE POWER OF M
IS EQUAL TO A TO THE POWER OF N,
THEN M IS EQUAL TO N.
AGAIN, IF WE HAVE EXPONENTIAL
EXPRESSIONS EQUAL TO EACH OTHER
WITH THE SAME BASE,
THE EXPONENTS MUST BE EQUAL.
SO, IN OUR FIRST EXAMPLE,
THE FIRST STEP IS TO ISOLATE 5
RAISED TO THE POWER OF X.
AND SINCE THIS 7 IS CONNECTED
BY MULTIPLICATION,
WE'LL DIVIDE BOTH SIDES BY 7,
7 OVER 7 SIMPLIFIES TO 1.
SO, WE HAVE 5 RAISED
TO THE POWER OF X EQUALS,
875 DIVIDED 7 IS EQUAL TO 125.
NOW THAT WE'VE ISOLATED THE
EXPONENTIAL PART ON ONE SIDE,
WE WANT TO TRY TO WRITE 125
WITH A BASE OF 5.
AND SINCE 125
IS EQUAL TO 5 TIMES 25
AND 25 IS EQUAL TO 5 TIMES 5,
125 IS EQUAL TO THREE
FACTORS OF 5,
WHICH WE CAN WRITE
AS 5 TO THE 3RD.
SO, WE HAVE 5 TO THE POWER
OF X EQUALS 5 TO THE 3RD.
NOW THAT THESE ARE EQUAL
AND THE BASES ARE THE SAME,
THE EXPONENTS MUST EQUAL
EACH OTHER,
THEREFORE X IS EQUAL TO 3.
IN OUR SECOND EQUATION,
WE WANT TO ISOLATE 3 RAISED
TO THE POWER OF X.
SO, WE NEED TO UNDO
THE SUBTRACTION
AND UNDO THIS MULTIPLICATION.
SO, WE'LL START BY ADDING 32
TO BOTH SIDES OF THE EQUATION.
THAT WILL BE ZERO,
SO WE HAVE 6 TIMES 3
TO THE POWER OF X EQUALS--
THIS WOULD BE 1458 AND NOW,
WE'LL DIVIDE BOTH SIDES BY 6.
OKAY AND THIS SIMPLIFIES TO 1
SO, WE HAVE 3 RAISED TO THE
POWER OF X EQUALS 1458
DIVIDED BY 6 IS EQUAL TO 243.
NOW, WE WANT TO SEE IF WE CAN
WRITE 243 WITH A BASE OF 3.
SO, 243 IS EQUAL TO--
WITH A NUMBER LIKE THIS,
IT'S HELPFUL TO REMEMBER
THAT IF THE SUM OF THE DIGITS
IS DIVISIBLE BY 3,
SO IS THE NUMBER.
AND SINCE 2 PLUS 4 PLUS 3
IS EQUAL TO 9,
WHICH IS DIVISIBLE BY 3,
SO IS THIS NUMBER.
SO, IT'S GOING TO BE
3 TIMES SOME NUMBER.
IT'S ACTUALLY 3 TIMES 81,
81 IS EQUAL TO 9 TIMES 9 AND
EACH 9 IS EQUAL TO 3 TIMES 3.
SINCE 243 IS EQUAL
TO 1, 2, 3, 4, 5 FACTORS OF 3,
WE CAN WRITE THIS AS 3 TO THE
POWER OF X EQUALS 3 TO THE 5TH.
NOW IF THESE ARE EQUAL
AND WE HAVE A COMMON BASE OF 3,
WE KNOW THE EXPONENTS
MUST BE EQUAL,
THEREFORE X IS EQUAL TO 5.
OKAY, WE'LL LOOK AT TWO MORE
EXAMPLES IN THE NEXT VIDEO.
