Warm Up

I can solve exponential equations using properties of logarithms

1. Solve for x: Round to the nearest hundredth

a) 𝟏𝟎𝒙 = 𝟏𝟑𝟎 b) 𝒍𝒐𝒈𝟕𝒙 = 𝟑 c) −𝟔 ∙ 𝟓𝒙 + 𝟏 = −𝟑𝟓

2. Expand: 𝒍𝒐𝒈𝟑 𝟔𝒙𝟐

3. Condense: 𝒍𝒐𝒈𝟒𝒚 + 𝒍𝒐𝒈𝟒𝟗 − 𝒍𝒐𝒈𝟒𝒛

Warm Up Solve for x:

a) 𝟏𝟎𝒙 = 𝟏𝟑𝟎 x = 2.11

b) 𝒍𝒐𝒈𝟕𝒙 = 𝟑 x = 343

Warm Up Solve for x:

c) −𝟔 ∙ 𝟓𝒙 + 𝟏 = −𝟑𝟓 X = 1.11

Warm Up

Expand: 𝒍𝒐𝒈𝟑 𝟔𝒙𝟐

𝒍𝒐𝒈𝟑 𝟔 + 𝟐𝒍𝒐𝒈𝟑(𝒙)

Condense: 𝒍𝒐𝒈𝟒𝒚 + 𝒍𝒐𝒈𝟒𝟗 − 𝒍𝒐𝒈𝟒𝒛

𝒍𝒐𝒈𝟒

𝟗𝒚

𝒛

Homework Questions

https://www.youtube.com/watch?v=N-7tcTIrers&safe=active

Quick Write On a separate sheet of paper, write for 5 minutes about your thoughts on the video. You might answer:

- Was this helpful?

- Do you have a better understanding of logarithms?

- What is a logarithm exactly?

Lets talk about math

• Successful mathematicians talk to each other.

• Risk going public with their ideas

• Defend, justify, critique, and revise their own ideas and the ideas of others

• I don’t know…YET! believe you can learn all of this material, before the test.

• Be kind to your classmates. Challenge the math not the mathematician.

High school redesign ideas

• You have about 8 minutes

• This is for points

• Money and space are unlimited

• The equipment/technology must currently exist in some form:

– for example: no teleportation rooms or zero-gravity game rooms in the new gym.

CCSS mathematical practice: MODELING

• Mathematical models are use to:

– Explain past phenomena or behavior

– Predict the future

– Test ideas

CCSS mathematical practice: MODELING

• 1: What is the question?

• 2. what facts do you need to answer the questions?

• 3. gather those facts and build the model – In math: the equation is the model

• 4. Use your model to answer the question.

• 5. confirm the accuracy of your model

• 6. Revise your model to improve accuracy as necessary.

What questions come to mind?

How many dominoes?

• How many dominoes would you have to line up, so that the last domino is as tall as a portland sky scraper?

• 1st: pick a number that is definitely too large • 2nd pick a number that is definitely too low. • 3rd take your best guess • 4th what do you need to know to make a

more accurate guess?

The worlds tallest buildings

• How many dominoes would you have to line up, so that the last domino is as tall as the worlds largest skyscraper?

• 1st: pick a number that is definitely to large

• 2nd pick a number that is definitely too low.

• 3rd take your best guess

• 4th what do you need to know to make a more accurate guess?

US Bancorp building: aka Big Pink

Big Pink- corners Big Pink from the east side.

Wells Fargo Building

The Empire State Building- fun fact, it was built in 1 year.

Dominoes!

Height of skyscrapers

• In Portland: that is over 160 meters (540-ish feet)

• The worlds tallest buildings are over 400m.

– (over 1200 feet), some double that.

• Now lets do the math to see if our guess was correct:

Producing a model

• The smallest domino was 1 x 5 mm.

• The dimensions of each domino increase by a scale factor of 1.5

• Write an exponential equation to model the height of each domino of the sequence, in meters.

• Remember the 1st domino is 5mm tall. (convert to meters)

Domino height

• Answer: giving height in meters

Y = .005(1.5) (x-1)

Solve the equation in for x: Y = .005(1.5) (x-1)

• Isolate the exponential:

𝑦

.005= 1.5𝑥−1

• convert to log form: X-1= log1.5(𝑦

.005)

• Solve for X: X= log1.5(𝑦

.005) + 1

• Where X is the position (1st, 3rd, 20th, ) of the domino with height of Y.

• Calculate X for 160 meters, and 400 meters • The 27th domino is the first over 160 meters • The 29 domino is the first that is more than 400

meters

The dominoes fall

Practice

15 minutes Check your answers on

the back table Anything not

finished is homework

Notes: Natural Log What is the approximate value of 𝜋?

What is the approximate value of e?

In math, e is known as Euler’s number. It has an approximate value of 2.718.

𝑙𝑜𝑔𝑒 is known as the “natural log” which is represented 𝑙𝑛

Notes: Natural Log Examples:

1) 𝑙𝑛10 =

2) ln 𝑥 = 4

3) 𝑒2𝑥−1 = 12

Notes: Solving for X

Ex. 1: log(x) + log(8) = 3

Step 1: Condense log(8x) = 3

Step 2: Rewrite 8x = 103

Step 3: Solve for x x = 125

Notes: Solving for X

• Ex 2: 𝑙𝑜𝑔4 𝑥 + 30 − 𝑙𝑜𝑔4 𝑥 = 3

Step 1: Condense 𝑙𝑜𝑔4𝑥+30

𝑥= 3

Step 2: Rewrite 𝑥+30

𝑥= 43

Step 3: Solve x = 0.48

Practice Solve each equation for x. Note: you may need to condense first!

1) 𝑙𝑜𝑔5 5𝑥 − 15 = 3

2) 2𝑙𝑜𝑔3𝑥 = 4

3) 𝑙𝑜𝑔2 4𝑥 − 12 − 3 = −1

4) 𝑙𝑜𝑔8𝑥 + 𝑙𝑜𝑔8 2𝑥 =4

5) 𝑙𝑜𝑔7 4𝑥 + 90 − 𝑙𝑜𝑔7𝑥 = 2

𝑥 = 28 𝑥 = 9 𝑥 = 4

𝑥 = 45.25 𝑥 = 2

Homework

• Please do the worksheet and STUDY FOR YOUR QUIZ!!!!!!!!