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
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.
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?
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)
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
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: 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