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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 logarithmic form.

Logarithmic Functions

Example: Write in logarithmic form.

Logarithmic Functions

Properties of Logarithms

Logarithmic Functions

Example: Expand logx12y

Logarithmic Functions

Properties of Logarithms

Logarithmic Functions

Example: Expand logx12/y

Logarithmic Functions

Properties of Logarithms

Logarithmic Functions

Properties of Logarithms

Logarithmic Functions

Properties of Logarithms

Logarithmic Functions

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

Logarithmic Functions

Example: Solve. log10= 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

- 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

=

≈ 2.34

Logarithmic Functions

Example: Solve. 9x-4 = 7.13

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. 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

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

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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