Logarithmic Functions TXVA: Not just a school, but an opportunity where every moment matters.

Objectives Convert between exponential and logarithmic expressions. Evaluate common and natural logarithms. Apply the rules of logarithms to rewrite expressions. Evaluate expressions that have uncommon bases using the change-of-base formula.

A logarithm is an Exponential function. All logarithms have a unique relationship with their exponential counterparts – they are actually the inverses of their exponential counterparts. log 3 729 =𝑦 can be written as 3 𝑦 =729. I use this to help me remember, “base to the answer power = the guts”. Try: log 4 256=𝑦

Use the rules of logarithms to expand: 𝑙𝑜𝑔 𝑥+3 4 𝑥−5 7 . Use the rule of logarithms to expand: ln 𝑎 3 𝑏 5 𝑐 8

Common Log & Natural Log A logarithmic function with base 10 is called a Common Log. Denoted: A logarithmic function with base e is called a Natural Log. *Note there is no base written.

Find y if 𝒚= 𝐥𝐨𝐠 𝟑𝟐 𝟏 . Explain your thinking.

Find y if 𝒚= 𝐥𝐨𝐠 𝟑𝟐 𝟏 . Explain your thinking. If log 𝑥=2 , what is 𝑥?

Write the expression 3 log 𝑥−2 + log 𝑥+2 − log (𝑥 2 −4) as a single logarithm. Write the expression5 ln 𝑀 +3 ln 𝑁 −2 ln 𝑃 − ln 𝑅 as a single logarithm.

Solve for x if 5= log 3 𝑥 2 +18 . Solve for x if −2= log 3𝑥+5 .

True or False? log 𝑥 + log 𝑦= log 𝑥𝑦 log 𝑥 − log 𝑦 =log⁡(𝑥−𝑦) ln 𝑥+𝑦 − ln 𝑥+𝑦 =0 log 𝑥 2 =2 log 𝑥

Write in exponential form: 𝑦= log 𝑥 Solve: log 4 1024=

Rewrite the following expression using a single logarithm. 2log 𝑥+3 +3 log 𝑥−7 − 5log 𝑥−2 + 2log (𝑥)

Solve: If , what is the value of ln 81? Let log 𝑥 𝐴=7; log 𝑥 𝐵=3; log 𝑥 𝐶=4 , what is the value of log 𝑥 𝐴 4 𝐶 3 𝐵 5 =?

Rewrite the following expression as a single logarithm: 2log 𝑥−1 +4 log 𝑥−2 −5 log 𝑥+1 −3 log 𝑥 Complete the sentence log 81 is the power to which ___________must be raised to produce a value of 81.

If log 𝐶 𝐷 =3 and log 𝐴 𝐷 =6 , what can you say about the relationship between A and C?

If log 𝑥=2 , what is 𝑥? Solve for 𝑥 : log 𝑥+2 + log 𝑥+3 = log 10𝑥

Solve for 𝑥 : log 𝑥 + log 𝑥−3 = log 3𝑥

.5801

-11

P=1000M

X = 6

E

The domain of the logarithmic function is The inverse function of an exponential function is the logarithmic function. For all positive real numbers x and b, b>0 and b1, y=logbx if and only if x=by. The domain of the logarithmic function is The range of the function is Since the log function is the inverse of the exponential function, their graphs are symmetric with respect to the line y=x.

**Remember that since the log function is the inverse of the exponential function, we can simply swap the x and y values of our important points! x y

There is a vertical asymptote at _______. Logarithmic Functions where b>1 are ___________, one-to-one functions. Logarithmic Functions where 0<b<1 are __________, one-to-one functions. The parent form of the graph has an x-intercept at (1,0) and passes through _____and _______ There is a vertical asymptote at _______. The value of b determines the flatness of the curve. The function is neither even nor odd. There is ______ symmetry. There is _____ local extrema. Increasing Decreasing (b,1) (1/b, -1) X=0 No no

More Characteristics of The domain is The range is End Behavior: As The x-intercept is The vertical asymptote is There is no y-intercept. There are no horizontal asymptotes. This is a ___________, __________ function. It is concave _________. (0,∞) (- ∞, ∞) As x ->0+, f(x)->- ∞ As x -> ∞, f(x)-> ∞ (1,0) X=0 Increasing, continuous down

Graph: Important Points: End Behavior: Domain: Range: x-intercept: Vertical Asymptote: x y Inc/dec? Concavity? End Behavior: (0, ∞) ( - ∞, ∞) (1, 0) X=0 Increasing Down As x ->0+, f(x)->- ∞ As x -> ∞, f(x)-> ∞

Graph: *Reflects @ x-axis. Important Points: Domain: Range: x-intercept: x y Vertical Asymptote: Inc/dec? Concavity? (0, ∞) ( - ∞, ∞) (1, 0) X=0 Decreasing Up As x ->0+, f(x)-> ∞ As x -> ∞, f(x)-> - ∞

Transformations Horizontal Reflect * Reflect Vertical Vertical. Domain: Range: x-intercept: Domain: Range: x-intercept: Domain: Range: x-intercept: Vertical Asymptote: Vertical Asymptote: Vertical Asymptote: Inc/dec? Inc/dec? Inc/dec? Concavity? Concavity? Concavity?

The asymptote of a logarithmic function of this form is the line To find an x-intercept in this form, let y=o in the equation To find a y-intercept in this form, let x=o in Since this is not possible, there is No y-intercept. is the vertical asymptote. is the x-intercept

Check it out!    Horizontal Horizontal is the vertical asymptote. is the x-intercept. Since this is not possible, there is No y-intercept.  Domain: Range: x-intercept: Vertical Asymptote: Inc/dec? Concavity?

Transformations Common Log Let to find the V.A. Horizontal. Horizontal is the V.A. Let to find the x-intercept. is the x-intercept. Domain: Range: Inc/Dec: Concavity: Let to find the y-intercept. is the y-intercept.

Transformations Natural Log Let to find the V.A. Reflect. Vertical. Horizontal Horizontal. Vertical is the V.A. Let to find the x-intercept. is the x-intercept. Let to find the y-intercept. Domain: Range: x-intercept: Inc/dec? Concavity? is the y-intercept.

Change of Base Formula Use this formula for entering logs with bases other than 10 or e in your graphing calculator. So, if you wanted to graph , you would enter in your calculator. Either the natural or common log may be used in the change of base formula. So, you could also enter in Your calculator.

Evaluate log 5 85 . Evaluate log 8 72 .

log 3 243 =________ If log 𝑥 =−5, what is x?

Tell me what you know….

Tell me what you know…. Domain: Range: Domain: Range:

5x

6+2 log x

Ln(x+3) - 4

X = -3

The domain is (-1, ∞)

X = ln(3)/7

B and C

A

10^ -2/3 +5

Ln(2) / ln (1.02) +2000 “C”