Logarithmic Model

class: center, middle, inverse, title-slide

.title[
# Logarithmic Model
]
.subtitle[
## Session 08
]
.author[
### Alejandro Ucan
]
.date[
### 2023-10-11
]

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# Goals

- Understand the logarithmic model.
- Solve logarithmic equations.
- State the criterium for logarithmic models.
- Use the logarithmic model to provide information about business problems.

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# Motivation:

> The sell of avocado during the days of the Super Bowl Season in the United States has an exponential behavior. In 2018, the sell of avocado was 105 million pounds, in 2019 was 115 million pounds, and in 2020 was 125.95 million pounds. How many years have to pass such that the sells be 200 million pounds of avocado?

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# Logarithmic Model

> __Definition:__ A logarithmic model is a model of the form `$y = c + b \log_a(x)$`, where `$a$` and `$b$` are constants where: <br/><br/>
  * - `$a > 0$` and `$a \neq 1,$` and it is known as the base of the logarithm. <br/><br/>
  * - `$b \neq 0,$` known as the scaling factor.<br/><br/>
  * - `$c$` is the vertical shift.

<br/><br/>
The logarithmic function is the inverse of the exponential function, we have the following equations:<br/><br/> `$$\log_a(y^x) = x \quadn a^{\log_a x) = x.$$`

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# Solve Logarithmic equations

Imagine that we want to know the value of `$x$` such that the following identy be true `$$4^x+3=19.$$`

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<br/><br/>
We can solve this equation by using the logarithmic function, `$$4^x=19-3=16.$$` therefore `$$x=\log_4 16=2.$$`

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# Solve Logarithmic equations

Imagine that we want to know the value of `$x$` such that the following identy be true `$$\log_2(x+1)=3.$$`

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<br/><br/>
We can solve this equation by using the exponential function, `$$2^3=x+1.$$` therefore `$$x=8-1=7.$$`

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# Solve Logarithmic equations

Imagine that we want to know the value of `$x$` such that the following identy be true `$$\log_2(x+1)=\log_2(2x-1).$$`

Thinks get complicated when we use more than one logarithm, but we can ease our equations using the following identities:

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## Logarithmic identities

- `$\log_a(x)+\log_a(y)=\log_a(xy)$`
- `$\log_a(x)-\log_a(y)=\log_a\left(\frac{x}{y}\right)$` 
- `$\log_a(x^r)=r\log_a(x)$`
- `$\log_a(x)=\frac{\log_b(x)}{\log_b(a)}$`
- `$a^{x+y}=a^x\cdot a^y$`
- `$a^{x-y}=\frac{a^x}{a^y}$`
- `$a^{x^r}=(a^x)^r$`
- `$a^x=\left(e^{\ln(a)}\right)^x=e^{x\ln(a)}$`

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# Example

Solve the following equation `$$\log_2(x+1)-\log_2(2x-1)=0.$$`

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<br/><br/>
We can use the logarithmic identities to solve this equation, `$$\log_2\left(\frac{x+1}{2x-1}\right)=0,$$` then we can apply exponential functions to both sides `$$2^0=\frac{x+1}{2x-1},$$` therefore `$$1=\frac{x+1}{2x-1},$$` and finally `$$x=2.$$`

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# Example

The supply and demand function for a manufacturer's product are `$$p=3^q\quad p=10e^{-q}$$` where price is measured in hundreds of pesos and quantity are measured in hundreds of units. Find the equilibrium price.

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<br/><br/>
The equilibrium of this function is when `$$3^q=10e^{-q},$$` therefore `$$q\ln(3)=\ln(10)-q,$$` and finally `$$q=\frac{\ln(10)}{\ln(3)+1}.$$`

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# Example

The supply and demand function for a manufacturer's product are `$$p=5^{q-2}\quad p=4e^{-q}$$` where price is measured in hundreds of pesos and quantity are measured in hundreds of units. Find the equilibrium price.

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<br/><br/>
The equilibrium of this function is when `$$5^{q-2}=4e^{-q},$$` therefore `$$(q-2)\ln(5)=\ln(4)-q,$$` and finally `$$q=\frac{\ln(4)+2\ln(5)}{\ln(5)+1}.$$`

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# Real world data

> We say that a real worl data `$(x,y)$` fits a logarithmic model if the behavior of the mean changes is decreasing or increasing always when the observations get too big.

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<br/><br/>
Consider the following data `$x=$` 1,2,3,4,5 and `$y=$` 0, 150.51, 238.56, 301.02, 349.48.

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