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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 Functions. Logarithmic Functions. The logarithmic function y = log a x, where a > 0 and a ≠ 1,

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Logarithmic Functions

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PreCalculusNYOS Charter SchoolQuarter 4“If we did all the things we were capable of doing, we would literally astound ourselves.” ~ Thomas Edison

Logarithmic Functions


Logarithmic Functions

  • The logarithmic function y = loga x,

    where a > 0 and a ≠ 1,

    is the inverseof the exponential function y = ax.

    y = loga x iff x = ay


Logarithmic Functions

Example: Write in exponential form.

log3 9 = 2


Logarithmic Functions

Example: Write in exponential form.

log3 9 = 2


Logarithmic Functions

Example: Write in exponential form.

log8 2 =


Logarithmic Functions

Example: Write in exponential form.

log8 2 =


Logarithmic Functions

Example: Write in exponential form.

log125 25 =


Logarithmic Functions

Example: Write in exponential form.

log125 25 =


Logarithmic Functions

Example: Write in logarithmic form.


Logarithmic Functions

Example: Write in logarithmic form.

log4 64 =


Logarithmic Functions

Example: Write in logarithmic form.


Logarithmic Functions

Example: Write in logarithmic form.

log3=


Logarithmic Functions

Example: Evaluate log7.

y = log7

y = -2


Logarithmic Functions

Example: Evaluate log5.

y = log5


Logarithmic Functions

Example: Evaluate log5.

y = log5

y = -3


Logarithmic Functions

Properties of Logarithms


Logarithmic Functions

Example: Expand log5 9x

= log5 9 + log5 x


Logarithmic Functions

Example: Expand logx12y


Logarithmic Functions

Example: Expand logx12y

= logx12 + logxy


Logarithmic Functions

Properties of Logarithms


Logarithmic Functions

Example: Expand log5 9/x

= log5 9 - log5 x


Logarithmic Functions

Example: Expand logx12/y


Logarithmic Functions

Example: Expand logx12/y

= logx12 - logxy


Logarithmic Functions

Properties of Logarithms


Logarithmic Functions

Example: Simplify log5 9x

= x log5 9


Logarithmic Functions

Properties of Logarithms


Logarithmic Functions

Example: Simplify. log5 9 = log5 x

9 = x


Logarithmic Functions

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

16 = 2x

8 = x


Logarithmic Functions

Properties of Logarithms


Logarithmic Functions

Example: Simplify. log5 5

1


Logarithmic Functions

Example: Simplify. log87 87

1


Logarithmic Functions

Example: Simplify. log87 1

0


Logarithmic Functions

Example: Simplify. log48 1

0


Logarithmic Functions

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


Logarithmic Functions

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

log8(48/w) = log86

48/w = 6

w = 8


Logarithmic Functions

Example: Solve. log10= x


Logarithmic Functions

Example: Solve. log10= x

log10= x

x =


Logarithmic Functions

  • 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:


Logarithmic Functions

Example: Rewrite with a base of 2.

log6 5

=


Logarithmic Functions

Example: Combine.

= log11 15


Logarithmic Functions

  • Natural logarithms have base e.

    ln 5


Logarithmic Functions

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

log6 254

=

≈ 3.09


Logarithmic Functions

Example: Convert log5 43 to a natural logarithm and evaluate.

log5 43


Logarithmic Functions

Example: Convert log5 43 to a natural logarithm and evaluate.

log5 43

=

≈ 2.34


Logarithmic Functions

Example: Solve using natural logs. 2x = 27

log2 27 = x

= x

x ≈ 4.75


Logarithmic Functions

Example: Solve. 9x-4 = 7.13


Logarithmic Functions

Example: Solve. 9x-4 = 7.13

log9 7.13 = x - 4

+ 4 = x

≈ 4.89


Logarithmic Functions

Example: Solve. 6x+2 = 14

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

ln6x+2 = ln 14


Logarithmic Functions

Example: Solve. 6x+2 = 14

ln6x+2 = ln 14

(x + 2) ln 6 = ln 14

x + 2 =

x ≈ -.53


Logarithmic Functions

Example: Solve. 2x-5 = 11


Logarithmic Functions

Example: Solve. 2x-5 = 11

ln2x-5= ln 11

(x – 5) ln 2 = ln11

x – 5 =

X ≈ 8.46


Logarithmic Functions

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


Logarithmic Functions

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

ln6x+2= ln14x-3

Move the exponents to the front and distribute…

x ln 6 + 2 ln6 = x ln 14 – 3 ln 14

Get the x terms on the left side and constants on the right…

x ln 6 - x ln 14 = – 3 ln14 – 2 ln 6

Factor out an x from the left side…

x (ln6 - ln 14) = – 3 ln 14 – 2 ln 6


Logarithmic Functions

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

x (ln 6 - ln 14) = – 3 ln 14 – 2 ln 6

x ≈ 13.57


Logarithmic Functions

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


Logarithmic Functions

Why ?

N = N0ekt


Logarithmic Functions

Why ?

N = N0ekt

2N0= N0ekt

2 = ekt

ln 2 = lnekt

ln 2 = kt


Logarithmic Functions

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?


Logarithmic Functions

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


Logarithmic Functions

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


Logarithmic Functions

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

k ≈ 17.33%


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