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- WE WANT TO EVALUATE THE TWO GIVEN LOGARITHMS
WITHOUT THE USE OF A CALCULATOR.
WE HAVE LOG BASE 3 OF 81 AND LOG BASE 2 OF 32.
TO EVALUATE THESE WE'RE GOING TO SET THEM EQUAL TO A VARIABLE,
LET'S SAY X,
THEN WE'LL WRITE THIS AS AN EXPONENTIAL EQUATION
TO DETERMINE THE VALUE OF X.
SO TO WRITE THIS LOG EQUATION AS AN EXPONENTIAL EQUATION
WE CAN USE OUR NOTES BELOW WHERE B IS THE BASE,
"A" IS THE EXPONENT, AND N IS THE NUMBER.
ANOTHER NICE WAY TO REMEMBER THIS IS TO START WITH THE BASE,
WORK AROUND THE EQUAL SIGN TO FORM THE EXPONENTIAL EQUATION.
SO HERE WE HAVE 3 RAISED TO THE POWER OF X MUST EQUAL 81.
SO 3 TO THE POWER OF X MUST EQUAL 81
AND NOW WE'LL SOLVE FOR X.
WE CAN DO THIS WITHOUT THE USE OF A CALCULATOR
BECAUSE WE CAN WRITE 81 AS 3 RAISED TO A POWER.
81 = 9 x 9 AND 9 = 3 x 3, SO 81 = 3 TO THE 4th.
SO NOW WE HAVE 3 TO THE X = 3 TO THE 4th.
SO THESE TWO ARE EQUAL AND THE BASES ARE THE SAME,
AND THEREFORE THE EXPONENTS MUST BE EQUAL
MEANING X MUST EQUAL 4.
WELL IF X = 4 THEN LOG BASE 3 OF 81 MUST = 4.
LET'S TAKE A LOOK AT A SECOND EXAMPLE.
WE'LL SET THIS EQUAL TO A VARIABLE LET'S SAY Y.
WRITE THIS AS AN EXPONENTIAL EQUATION.
SO 2 IS THE BASE, Y IS THE EXPONENT AND THE NUMBER IS 32.
SO 2 TO THE POWER OF Y MUST EQUAL 32.
LET'S TAKE A LOOK AT 32.
32 IS EQUAL TO 2 x 16, 16 IS EQUAL TO 4 x 4,
4 IS EQUAL 2 x 2.
SO WE HAVE 1, 2, 3, 4, 5 FACTORS OF 2 SO 32 IS 2 TO THE 5th.
SO 2 TO THE POWER OF Y EQUALS 2 TO THE 5th.
AND AGAIN, THESE ARE EQUAL. THE BASES ARE THE SAME.
SO THE EXPONENTS MUST BE EQUAL AND THEREFORE Y IS = TO 5.
WHICH MEANS OUR LOGARITHM IS = TO 5.
SO WE HAVE LOG BASE 2 OF 32 = 5.
NEXT, WE'LL TAKE A LOOK AT TWO EXAMPLES
WHEN THE NUMBER PART OF THE LOGARITHM IS A FRACTION.