Exponentials
An exponential function (graph) is one where the variable (x) is a power.
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The general formula for these is...
$$ y = a^{x} $$ 
The power will be equal to 0 when x is 2. In this instance, the y coordinate would then be 0 as -(1)+1 = 0 Due to the negative coefficient of x, the graph decreases exponentially and so the larger x gets, the smaller 3^{6-3x} gets and as that approaches 0, the closer f(x) gets to 1 
In terms of functions, we can consider this graph a transformation of our graph from earlier.
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$$ f(x) = 2^{x} $$
For any exponential function, if the power is 0, the value is 1 as any number to the power of 0 is equal to 1. This means, when x=0 on this graph (at the y axis), y = 1 In this graph, as x (the power) increases, the value of y increases at an increasing rate (exponentially). As x decreases, the value of y also decreases and the smaller the value of x, the closer y is to 0. y approaches 0 but does not reach 0 and so gets smaller asymptotically. You can test these theories by imputing different values of x The graph is shown to the left. $$ f(x) = -3^{6-3x}+1 $$
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All the examples above have a=2, but a also has an impact on the graph...
As a gets larger (if a>1), y increases as x increases. At the sae time the difference between a with one power and a with the next will become greater. For example, the difference between 2 with power 1 and power 2 is (4-1=) 3; but 3 to the power of 2 minus 3 to the power of 1 is (9-3=) 6. This suggests that as a gets larger the rate of increase increases, and the same happens with the rate of increase. This means as a increases the gradient change of the exponential graph, y, increases so the curve gets steeper.
It is worth noting that fractions to the power of x, like 1/a, can also be written as a to the power of -x...
$$ (\frac {1}{a})^{x} = a^{-x} $$
As a gets larger (if a>1), y increases as x increases. At the sae time the difference between a with one power and a with the next will become greater. For example, the difference between 2 with power 1 and power 2 is (4-1=) 3; but 3 to the power of 2 minus 3 to the power of 1 is (9-3=) 6. This suggests that as a gets larger the rate of increase increases, and the same happens with the rate of increase. This means as a increases the gradient change of the exponential graph, y, increases so the curve gets steeper.
It is worth noting that fractions to the power of x, like 1/a, can also be written as a to the power of -x...
$$ (\frac {1}{a})^{x} = a^{-x} $$
The Graphs on this page where produced with the use of Desmos. We would definitely recommend playing around with this to consolidate your understanding of graphs.
The term 'exponential' is probably most associated with exponential growth and exponential decay where a model, system or situation can be described as having an increase or decrease at an increasing rate.
Exponential Growth - the rate of growth increases as amount increases
Exponential Decay - the rate of decay increases as amount decreases
Such situations are often modelled as...
$$ y = k^{x} $$
where k is the base number and x is the independent variable. (K stays the same while x varies to produce different values of y). There is often also a constant added or multiplied to complete the model.
For example, a student's test scores increases exponentially as the number of Scigo pages they read increases, in a way modelled as follows...
$$ y = a*1.1^{x} $$
where y is the test score, a is the test score the student would get without reading any scigo pages, and x is the number of pages they read. Note how, when x=0, 1.1 to the power of x is 1, so their test score is a (hence the test score without reading any scigo pages).
The base number here is 1.1 while the independent variable is x, and as x grows larger, so does y, so this is exponential growth.
Exponential Growth - the rate of growth increases as amount increases
Exponential Decay - the rate of decay increases as amount decreases
Such situations are often modelled as...
$$ y = k^{x} $$
where k is the base number and x is the independent variable. (K stays the same while x varies to produce different values of y). There is often also a constant added or multiplied to complete the model.
For example, a student's test scores increases exponentially as the number of Scigo pages they read increases, in a way modelled as follows...
$$ y = a*1.1^{x} $$
where y is the test score, a is the test score the student would get without reading any scigo pages, and x is the number of pages they read. Note how, when x=0, 1.1 to the power of x is 1, so their test score is a (hence the test score without reading any scigo pages).
The base number here is 1.1 while the independent variable is x, and as x grows larger, so does y, so this is exponential growth.
Above is the graph of this model with the higher, blue line representing a student with a=60, and the lower red line representing a student where a=30. It's worth noticing that the only relevant parts are those with positive values of x as you cannot read a negative number of pages. With this test, the maximum mark is also 100 so anything above this in the y axis is also not relevant.
Therefore, we can use the graph to determine the student who would've gotten 30% without scigo, should read 13 pages to get 100% while the student set for 60% should read 6. Note how we rounded up each time. We did this as it's easier to read a whole page than to guess you've read a specific fraction of a page. It is also worth noting that in reality, there are likely to be many other factors that govern test performance, which could suggest thus model is limited.
Therefore, we can use the graph to determine the student who would've gotten 30% without scigo, should read 13 pages to get 100% while the student set for 60% should read 6. Note how we rounded up each time. We did this as it's easier to read a whole page than to guess you've read a specific fraction of a page. It is also worth noting that in reality, there are likely to be many other factors that govern test performance, which could suggest thus model is limited.
In another example, the amount of uranium left in a sample after t years can be expressed by the following equation...
$$ m = 1000*12^{-0.001t} $$
Here, m represents the mass of uranium while t is the independent variable and represents time. 12 is the base number. The initial mass of uranium (mass when time is 0, so t=0) is 1000. As t gets larger, m gets smaller given the exponent is negative. This is therefore exponential decay.
$$ m = 1000*12^{-0.001t} $$
Here, m represents the mass of uranium while t is the independent variable and represents time. 12 is the base number. The initial mass of uranium (mass when time is 0, so t=0) is 1000. As t gets larger, m gets smaller given the exponent is negative. This is therefore exponential decay.
Here the x axis should be called the t axis while the y axis is instead the m axis. Again, we are not interested in the time before t=0 so only care about when t is positive. In this graph, you'll also find that m=0 is an asymptote suggesting the amount of uranium can never reach 0. The half life of a radioactive substance is the mount of time it takes for the substance to be halved in mass (half having decayed), which can be read at 500 on this graph from the m axis. Thus suggests the half life is around 280 years.
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There is a number that is frequently seen as the base number with exponentials and that number is, Euler's Number, or just e. This number is a remarkable irrational number found frequently in nature and calculated by many different methods. One of these is shown on the right, another is the limit (asymptote) of (1+1/n)^n.
To 2 decimal places, e is 2.71, and most relevantly... $$ the gradient of the graph of e^{x} is e^{x} $$ |
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e is just a number even if irrational, so e^x is just an exponential graph which you can see with its shape to the left. It also passes through (0,1) with an asymptote of y=0 and can be transformed like any other exponential.
However, this graph can also represent the gradient function or derivative of e^x. This is called the natural exponential function or just the 'exponential function.' |
Below (on the left) are the rules for differentiating graphs with e as part of the base number, and (to the right) a demonstration of chain rule which is how the rules were derived.
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Note that the gradient of Ae^kx is directly proportional to itself. This means it can be used to model situations where the rate of change of y is proportional to y...
Logarithms
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Hi! You're still here? Cool! Let me log that down...
A logarithm is the inverse of an exponential. With an exponential, you can work out the value of another after it has been raised to an exponent (a power). With a logarithm, you have the initial number and the number after it's been raised to an exponent, and the logarithm will work out what the exponent was. The right hand picture depicts both the logarithm and its complementary exponential. Note how the logarithm is represented by log with the base as the subscript (the blue b). |
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$$ log_{ 3} 9 $$
This is asking "what power does 3 need to be raised to, to equal 9", the answer is 2 so... $$ log_{ 3} 9=2 $$ and... $$ 3^{2}=9 $$ |
$$ log_{ 4} 2 $$
This is asking "what power does 4 need to be raised to to equal 2", the answer is 0.5 so... $$ log_{ 4} 2 = \frac {1}{2} $$ and... $$ 4^{\frac {1}{2}} = 2 $$ |
HOLD THE PHONE...
I won't log the bill for that (please forgive me, I think I've snuck this past my boss)
There are certain cases where logarithms don't work...
There are also some specific logarithms you need to know...
I won't log the bill for that (please forgive me, I think I've snuck this past my boss)
There are certain cases where logarithms don't work...
- Logarithms don't accept a negative base number
- These would only work with some numbers as many examples, such as log with base -3 of 3, are impossible
- These would only work with some numbers as many examples, such as log with base -3 of 3, are impossible
- Logarithms don't accept a negative argument
- These would only work with some numbers as many examples, such as log with base 3 of -3, are impossible
- These would only work with some numbers as many examples, such as log with base 3 of -3, are impossible
- Logarithms don't accept a base nor argument of 0
- 0 to any power could only ever be 0*, there is therefore no way of knowing what power it is to, and any other argument makes no sense. 0 also can't be the result of any other power. *Even 0^0 is undefined because it could either be 1 (every number to the power of 0 is 1) or 0 (0 multiplied by any number is 0.
- Logarithms don't accept a base of 1
- 1 to any power could only ever be 1. There is no way of knowing what power it is to.
There are also some specific logarithms you need to know...
- The Common Logarithm is a logarithm with base 10 and is often written as just log a
- The Natural Logarithm is a logarithm with base e and is often written as just ln a (we discuss e in the exponential section)
To demonstrate that logarithms and exponentials are inverses we can graph them with the same base...
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We can see in both cases that the logarithm (without transformation) has an asymptote at x=0 and passes through (1,0)
You may notice that this is just like the exponential except y=0 has become x=0 and (0,1) has become (1,0). In other words the x and y have switched which is why logarithms are considered the inverse of exponentials. (x, the number that goes into the function, has switched with y, the number that comes out)
This makes sense because with 10^x, x is the exponent and is going in, and the argument comes out. Meanwhile, with log x, x is the argument and is going in, and the exponent comes out.
The dashed black line is along y=x as the inverse of a function is its reflection in the line y=x.
You may notice that this is just like the exponential except y=0 has become x=0 and (0,1) has become (1,0). In other words the x and y have switched which is why logarithms are considered the inverse of exponentials. (x, the number that goes into the function, has switched with y, the number that comes out)
This makes sense because with 10^x, x is the exponent and is going in, and the argument comes out. Meanwhile, with log x, x is the argument and is going in, and the exponent comes out.
The dashed black line is along y=x as the inverse of a function is its reflection in the line y=x.
Logarithm Rules
As exponentials have rules, so do logarithms. The rules for exponentials and logarithms both are tabled below...
With an understanding of logarithms, we can now dive further into modelling with exponentials (and logarithms).