- 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.
