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Section 6-6 Solving Exponential Equations
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Section 6-6Solving Exponential Equations
Warm-upGIVE FIVE PAIRS (X, Y) OF NUMBERS FOR WHICH YOU CAN DETERMINE EXACT SOLUTIONS TO
THE EQUATION:
15y = x
Warm-upGIVE FIVE PAIRS (X, Y) OF NUMBERS FOR WHICH YOU CAN DETERMINE EXACT SOLUTIONS TO
THE EQUATION:
15y = x
HOW ABOUT (1, 0), (15, 1), (225, 2), (3375, 3), ETC.
Warm-upGIVE FIVE PAIRS (X, Y) OF NUMBERS FOR WHICH YOU CAN DETERMINE EXACT SOLUTIONS TO
THE EQUATION:
15y = x
HOW ABOUT (1, 0), (15, 1), (225, 2), (3375, 3), ETC.
ALL OF THESE ANSWERS ALSO FIT THE EQUATION:
y = log15
x
Exponential Equation
Exponential Equation
AN EQUATION THAT HAS A VARIABLE IN THE EXPONENT
Blog Question
Blog Question
HOW ARE EXPONENTIAL EQUATIONS AND LOGARITHMIC EQUATIONS RELATED?
Example 1IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
Example 1IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
Example 1IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
log2
3 = t
Example 1IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
log2
3 = t
WHAT ABOUT TAKING THE LOGARITHM OF EACH SIDE OF THE EXPONENTIAL?
Example 1IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
log2
3 = t
WHAT ABOUT TAKING THE LOGARITHM OF EACH SIDE OF THE EXPONENTIAL?
2t = 3
Example 1IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
log2
3 = t
WHAT ABOUT TAKING THE LOGARITHM OF EACH SIDE OF THE EXPONENTIAL?
2t = 3
log2t = log3
Example 1IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
log2
3 = t
WHAT ABOUT TAKING THE LOGARITHM OF EACH SIDE OF THE EXPONENTIAL?
2t = 3
log2t = log3
t log2 = log3
Example 1IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
log2
3 = t
WHAT ABOUT TAKING THE LOGARITHM OF EACH SIDE OF THE EXPONENTIAL?
2t = 3
log2t = log3
t log2 = log3 log2 log2
Example 1IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
log2
3 = t
WHAT ABOUT TAKING THE LOGARITHM OF EACH SIDE OF THE EXPONENTIAL?
2t = 3
log2t = log3
t log2 = log3 log2 log2
t = log3
log2
Example 1IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
log2
3 = t
WHAT ABOUT TAKING THE LOGARITHM OF EACH SIDE OF THE EXPONENTIAL?
2t = 3
log2t = log3
t log2 = log3 log2 log2
t = log3
log2 ≈ 1.58
Change of Base Theorem
Change of Base Theorem
FOR ALL VALUES OF a, b, AND c WHERE THE LOGS EXIST:
log
ba =
logc
a
logc
b=
lnalnb
Change of Base Theorem
FOR ALL VALUES OF a, b, AND c WHERE THE LOGS EXIST:
log
ba =
logc
a
logc
b=
lnalnb
THIS MEANS WE CAN SOLVE ANY LOGARITHM USING COMMON LOGS OR NATURAL LOGS
Example 2EVALUATE:
Example 2EVALUATE:
log17
34
Example 2EVALUATE:
log17
34
=
log34log17
Example 2EVALUATE:
log17
34
=
log34log17
≈ 1.24
Example 2EVALUATE:
log17
34 log17
34
=
log34log17
≈ 1.24
Example 2EVALUATE:
log17
34 log17
34
=
log34log17
≈ 1.24
=
ln34ln17
Example 2EVALUATE:
log17
34 log17
34
=
log34log17
≈ 1.24
=
ln34ln17
≈ 1.24
Example 3THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
Example 3THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
A(t ) = Pert
Example 3THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
300,000,000 = 250,000,000e .01t A(t ) = Pert
Example 3THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
300,000,000 = 250,000,000e .01t A(t ) = Pert
250,000,000 250,000,000
Example 3THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
300,000,000 = 250,000,000e .01t A(t ) = Pert
250,000,000 250,000,000 1.2 = e .01t
Example 3THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
300,000,000 = 250,000,000e .01t A(t ) = Pert
250,000,000 250,000,000 1.2 = e .01t
ln1.2 = lne .01t
Example 3THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
300,000,000 = 250,000,000e .01t A(t ) = Pert
250,000,000 250,000,000 1.2 = e .01t
ln1.2 = lne .01t
ln1.2 = .01t
Example 3THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
300,000,000 = 250,000,000e .01t A(t ) = Pert
250,000,000 250,000,000 1.2 = e .01t
ln1.2 = lne .01t
ln1.2 = .01t t =ln1.2.01
Example 3THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
300,000,000 = 250,000,000e .01t A(t ) = Pert
250,000,000 250,000,000 1.2 = e .01t
ln1.2 = lne .01t
ln1.2 = .01t t =ln1.2.01 t ≈ 18
Example 3THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
300,000,000 = 250,000,000e .01t A(t ) = Pert
250,000,000 250,000,000 1.2 = e .01t
ln1.2 = lne .01t
ln1.2 = .01t t =ln1.2.01 t ≈ 18 YEARS
Example 4IN SECTION 6-2, THE FORMULA T(p) = 10p0.585 IS GIVEN FOR THE
APPROXIMATE TIME T(p) IT TAKES TO MICROWAVE p PORTIONS OF FOOD, ASSUMING THAT DOUBLING THE NUMBER OF PORTIONS REQUIRES COOKING
TIME TO BE MULTIPLIED BY 1.5. THE FORMULA WAS SAID TO HAVE BEEN DEDUCED FROM THE FACT THAT 1 PORTION REQUIRED 10 MINUTES TO COOK AND 2 PORTIONS REQUIRED 15 MINUTES. SHOW HOW THIS FORMULA WAS CALCULATED FROM THE ASSUMPTION THAT THERE WAS A MODEL OF THE
FORM T(p) = 10px.
T (1) = 10i1x = 10 T (2) = 10i2x = 15
Example 4IN SECTION 6-2, THE FORMULA T(p) = 10p0.585 IS GIVEN FOR THE
APPROXIMATE TIME T(p) IT TAKES TO MICROWAVE p PORTIONS OF FOOD, ASSUMING THAT DOUBLING THE NUMBER OF PORTIONS REQUIRES COOKING
TIME TO BE MULTIPLIED BY 1.5. THE FORMULA WAS SAID TO HAVE BEEN DEDUCED FROM THE FACT THAT 1 PORTION REQUIRED 10 MINUTES TO COOK AND 2 PORTIONS REQUIRED 15 MINUTES. SHOW HOW THIS FORMULA WAS CALCULATED FROM THE ASSUMPTION THAT THERE WAS A MODEL OF THE
FORM T(p) = 10px.
T (1) = 10i1x = 10 T (2) = 10i2x = 15
1x = 1
Example 4IN SECTION 6-2, THE FORMULA T(p) = 10p0.585 IS GIVEN FOR THE
APPROXIMATE TIME T(p) IT TAKES TO MICROWAVE p PORTIONS OF FOOD, ASSUMING THAT DOUBLING THE NUMBER OF PORTIONS REQUIRES COOKING
TIME TO BE MULTIPLIED BY 1.5. THE FORMULA WAS SAID TO HAVE BEEN DEDUCED FROM THE FACT THAT 1 PORTION REQUIRED 10 MINUTES TO COOK AND 2 PORTIONS REQUIRED 15 MINUTES. SHOW HOW THIS FORMULA WAS CALCULATED FROM THE ASSUMPTION THAT THERE WAS A MODEL OF THE
FORM T(p) = 10px.
T (1) = 10i1x = 10 T (2) = 10i2x = 15
1x = 1
log1= 0
Example 4IN SECTION 6-2, THE FORMULA T(p) = 10p0.585 IS GIVEN FOR THE
APPROXIMATE TIME T(p) IT TAKES TO MICROWAVE p PORTIONS OF FOOD, ASSUMING THAT DOUBLING THE NUMBER OF PORTIONS REQUIRES COOKING
TIME TO BE MULTIPLIED BY 1.5. THE FORMULA WAS SAID TO HAVE BEEN DEDUCED FROM THE FACT THAT 1 PORTION REQUIRED 10 MINUTES TO COOK AND 2 PORTIONS REQUIRED 15 MINUTES. SHOW HOW THIS FORMULA WAS CALCULATED FROM THE ASSUMPTION THAT THERE WAS A MODEL OF THE
FORM T(p) = 10px.
T (1) = 10i1x = 10 T (2) = 10i2x = 15
1x = 1
log1= 0THIS WON’T HELP
Example 4IN SECTION 6-2, THE FORMULA T(p) = 10p0.585 IS GIVEN FOR THE
APPROXIMATE TIME T(p) IT TAKES TO MICROWAVE p PORTIONS OF FOOD, ASSUMING THAT DOUBLING THE NUMBER OF PORTIONS REQUIRES COOKING
TIME TO BE MULTIPLIED BY 1.5. THE FORMULA WAS SAID TO HAVE BEEN DEDUCED FROM THE FACT THAT 1 PORTION REQUIRED 10 MINUTES TO COOK AND 2 PORTIONS REQUIRED 15 MINUTES. SHOW HOW THIS FORMULA WAS CALCULATED FROM THE ASSUMPTION THAT THERE WAS A MODEL OF THE
FORM T(p) = 10px.
T (1) = 10i1x = 10 T (2) = 10i2x = 15
2x = 1.5 1
x = 1
log1= 0THIS WON’T HELP
Example 4IN SECTION 6-2, THE FORMULA T(p) = 10p0.585 IS GIVEN FOR THE
APPROXIMATE TIME T(p) IT TAKES TO MICROWAVE p PORTIONS OF FOOD, ASSUMING THAT DOUBLING THE NUMBER OF PORTIONS REQUIRES COOKING
TIME TO BE MULTIPLIED BY 1.5. THE FORMULA WAS SAID TO HAVE BEEN DEDUCED FROM THE FACT THAT 1 PORTION REQUIRED 10 MINUTES TO COOK AND 2 PORTIONS REQUIRED 15 MINUTES. SHOW HOW THIS FORMULA WAS CALCULATED FROM THE ASSUMPTION THAT THERE WAS A MODEL OF THE
FORM T(p) = 10px.
T (1) = 10i1x = 10 T (2) = 10i2x = 15
2x = 1.5
log21.5 = x
1x = 1
log1= 0THIS WON’T HELP
Example 4IN SECTION 6-2, THE FORMULA T(p) = 10p0.585 IS GIVEN FOR THE
APPROXIMATE TIME T(p) IT TAKES TO MICROWAVE p PORTIONS OF FOOD, ASSUMING THAT DOUBLING THE NUMBER OF PORTIONS REQUIRES COOKING
TIME TO BE MULTIPLIED BY 1.5. THE FORMULA WAS SAID TO HAVE BEEN DEDUCED FROM THE FACT THAT 1 PORTION REQUIRED 10 MINUTES TO COOK AND 2 PORTIONS REQUIRED 15 MINUTES. SHOW HOW THIS FORMULA WAS CALCULATED FROM THE ASSUMPTION THAT THERE WAS A MODEL OF THE
FORM T(p) = 10px.
T (1) = 10i1x = 10 T (2) = 10i2x = 15
2x = 1.5
log21.5 = x
=
log1.5log2
1x = 1
log1= 0THIS WON’T HELP
Example 4IN SECTION 6-2, THE FORMULA T(p) = 10p0.585 IS GIVEN FOR THE
APPROXIMATE TIME T(p) IT TAKES TO MICROWAVE p PORTIONS OF FOOD, ASSUMING THAT DOUBLING THE NUMBER OF PORTIONS REQUIRES COOKING
TIME TO BE MULTIPLIED BY 1.5. THE FORMULA WAS SAID TO HAVE BEEN DEDUCED FROM THE FACT THAT 1 PORTION REQUIRED 10 MINUTES TO COOK AND 2 PORTIONS REQUIRED 15 MINUTES. SHOW HOW THIS FORMULA WAS CALCULATED FROM THE ASSUMPTION THAT THERE WAS A MODEL OF THE
FORM T(p) = 10px.
T (1) = 10i1x = 10 T (2) = 10i2x = 15
2x = 1.5
log21.5 = x
=
log1.5log2 ≈ .585
1x = 1
log1= 0THIS WON’T HELP
Homework
Homework
P. 406 #1-20
