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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

- 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

Example: Write in exponential form.

log3 9 = 2

Example: Write in exponential form.

log3 9 = 2

Example: Write in exponential form.

log8 2 =

Example: Write in exponential form.

log8 2 =

Example: Write in exponential form.

log125 25 =

Example: Write in exponential form.

log125 25 =

Example: Write in logarithmic form.

Example: Write in logarithmic form.

log4 64 =

Example: Write in logarithmic form.

Example: Write in logarithmic form.

log3=

Example: Evaluate log7.

y = log7

y = -2

Example: Evaluate log5.

y = log5

Example: Evaluate log5.

y = log5

y = -3

Properties of Logarithms

Example: Expand log5 9x

= log5 9 + log5 x

Example: Expand logx12y

Example: Expand logx12y

= logx12 + logxy

Properties of Logarithms

Example: Expand log5 9/x

= log5 9 - log5 x

Example: Expand logx12/y

Example: Expand logx12/y

= logx12 - logxy

Properties of Logarithms

Example: Simplify log5 9x

= x log5 9

Properties of Logarithms

Example: Simplify. log5 9 = log5 x

9 = x

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

16 = 2x

8 = x

Properties of Logarithms

Example: Simplify. log5 5

1

Example: Simplify. log87 87

1

Example: Simplify. log87 1

0

Example: Simplify. log48 1

0

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

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

log8(48/w) = log86

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

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

log5 43

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

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.

ln6x+2 = ln 14

Example: Solve. 6x+2 = 14

ln6x+2 = ln 14

(x + 2) ln 6 = ln 14

x + 2 =

x ≈ -.53

Example: Solve. 2x-5 = 11

Example: Solve. 2x-5 = 11

ln2x-5= ln 11

(x – 5) ln 2 = ln11

x – 5 =

X ≈ 8.46

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

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

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

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 = lnekt

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%