PRECALCULUS NYOS CHARTER SCHOOL QUARTER 4 “IF WE DID ALL THE THINGS WE WERE CAPABLE OF DOING, WE WOULD LITERALLY ASTOUND OURSELVES.” ~ THOMAS EDISON Logarithmic

PRECALCULUSNYOS CHARTER SCHOOLQUARTER 4

“IF WE DID ALL THE THINGS WE WERE CAPABLE OF DOING, WE WOULD LITERALLY ASTOUND OURSELVES.” ~ THOMAS EDISON

Logarithmic Functions


The logarithmic function y = loga x,

where a > 0 and a ≠ 1,

is the inverse of the exponential function y = ax.

y = loga x iff x = ay


Example: Write in exponential form.

log3 9 = 2



log3 9 = 2



log8 2 =



log8 2 =



log125 25 =



log125 25 =


Example: Write in logarithmic form.



log4 64 =





log3 =


Example: Evaluate log7 .

y = log7

y = -2



y = log5



y = log5

y = -3


Properties of Logarithms

Property Definition

Product logb mn = logb m + logb n


Example: Expand log5 9x

= log5 9 + log5 x


Example: Expand logx 12y


Example: Expand logx 12y

= logx 12 + logx y



Property Definition


Quotient logb = logb m - logb n


Example: Expand log5 9/x

= log5 9 - log5 x


Example: Expand logx 12/y


Example: Expand logx 12/y

= logx 12 - logx y



Property Definition



Power logb mp = p logb m


Example: Simplify log5 9x

= x log5 9



Property Definition




EqualityIf logb m = logb n, then m =

n


Example: Simplify. log5 9 = log5 x

9 = x


Example: Solve for x. log5 16 = log5 2x

16 = 2x

8 = x



Property Definition




EqualityIf logb m = logb n, then m =

n

Identity loga a = 1

Zero loga 1 = 0


Example: Simplify. log5 5

1



1



0



0


Example: Solve. log8 48 – log8 w = log8 6


Example: Solve. log8 48 – log8 w = log8 6

log8 (48/w) = log8 6

48/w = 6

w = 8


Example: Solve. log10 = x


Example: Solve. log10 = x

log10 = x

x =


If a, b, and n are positive numbers and neither a nor b is 1, then the following is called the change of base formula:


Example: Rewrite with a base of 2.

log6 5

=


Example: Combine.

= log11 15


Natural logarithms have base e.

ln 5


Example: Convert log6 254 to a natural logarithm and evaluate.

log6 254

= ≈ 3.09



log5 43



log5 43

= ≈ 2.34


Example: Solve using natural logs. 2x = 27

log2 27 = x

= x

x ≈ 4.75


Example: Solve. 9x-4 = 7.13


Example: Solve. 9x-4 = 7.13

log9 7.13 = x - 4

+ 4 = x

≈ 4.89


Example: Solve. 6x+2 = 14

The variable is in the exponent. Take the log of both sides.

ln 6x+2 = ln 14


Example: Solve. 6x+2 = 14

ln 6x+2 = ln 14

(x + 2) ln 6 = ln 14

x + 2 =

x ≈ -.53


Example: Solve. 2x-5 = 11


Example: Solve. 2x-5 = 11

ln 2x-5 = ln 11

(x – 5) ln 2 = ln 11

x – 5 =

X ≈ 8.46


Example: Solve. 6x+2 = 14x-3



ln 6x+2 = ln 14x-3 Move the exponents to the front and distribute…

x ln 6 + 2 ln 6 = x ln 14 – 3 ln 14Get the x terms on the left side and constants on the

right…

x ln 6 - x ln 14 = – 3 ln 14 – 2 ln 6 Factor out an x from the left side…

x (ln 6 - ln 14) = – 3 ln 14 – 2 ln 6



x (ln 6 - ln 14) = – 3 ln 14 – 2 ln 6

x ≈ 13.57


Sometimes we may want to know how long it takes for a quantity modeled by an exponential function to double.


Why ?

N = N0ekt


Why ?

N = N0ekt

2N0 = N0ekt

2 = ekt ln 2 = ln ekt

ln 2 = kt


Example: As a freshman in college, McKayla received $4,000 from her great aunt. She invested the money and would like to buy a car that costs twice that amount when she graduates in four years. If the money is invested in an account that pays 9.5% compounded continuously, will she have enough money for the car?


Example: $4,000; 9.5%; double in 4 yrs?


Example: What interest rate is required for an amount to double in 4 years?


Example: What interest rate is required for an amount to double in 4 years?

k ≈ 17.33%

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PRECALCULUS NYOS CHARTER SCHOOL QUARTER 4 “IF WE DID ALL THE THINGS WE WERE CAPABLE OF DOING, WE WOULD LITERALLY ASTOUND OURSELVES.” ~ THOMAS EDISON Logarithmic