Exponentials and Logarithms
EXPONENTIALS AND LOGARITHMS
1. Exponential Functions
The graph of passes through (0, 1), since . The greater the value of a, the steeper the curve.
The graph of is a reflection of the graph of in the y-axis.
C2 p39 Ex 3A
2. Logarithms
Consider the identity
Here, the 3 is a power or index. We can rewrite the identity involving a logarithm.
We say that 3 is the logarithm, to base 2, of 8. So logarithm is simply another word for a power or index.
In the same way,
Note that the value of the logarithm depends upon the base. For example,
In general,
It is important to be able to freely convert between the index and logarithm form of an equation.
Logarithms to base 10 are very common, and the base is often omitted from the log. So , and , both mean .
C2 p40 Ex 3B, p41 Ex 3C
3. Law 1 :
Generalising,
On a calculator, log 10 = 1, as expected.
4. Law 2 : Logarithm of 1
Generalising,
On a calculator, log 1 = 0, as expected.
5. Law 3 : Log of a Product
Consider this product.
On a calculator
It seems that , so the log of a product is the same as the sum of the logs.
To show why, we first write our identities in index form.
Rewriting in logarithm form,
…and there you go.
The general proof follows the same line of reasoning. Let
Then
This proves the law…
6. Law 4 : Log of a Quotient
From the previous section,
This means that the log of a quotient can be found by subtracting the logs. To show why, we first write our identities in index form.
Rewriting in logarithm form,
The general proof is as follows.
This proves the law…
7. Law 5 : Log of a Power
Consider this identity.
Taking logs of both sides, and using our result for the log of a product
So it seems that the power can just drop down the front!Checking on a calculator,
Using our rule, we would expect
Checking on a calculator,
The general proof is as follows. Let
Then
And so we have our general rule…
A special case of this law is
A final calculator check...
Example1:Simplify.
Example 2 : Express in terms of , and.
Activity1:Given only that and , find the logarithms of as many numbers between 1 and 100 as possible without using a calculator. Use the Logarithms to 100worksheet for this.
C2 p43 Ex 3D
8. Exponential Equations
To solve
we simply take the cube root of both sides...
To solve
we need to use logarithms. We could rewrite the equation as...
...but this does not help, as calculators only give logarithms to base 10.
Instead we take logs (to base 10) of both sides.
We can check this answer by doing on a calculator, and obtaining 4.
Example1: Solve the equation .
Example 2 : Solve the equation .
Let,
Therefore
and
This example illustrates the need to be on the look out for ‘quadratic equations in disguise’. The next example is even more disguised.
Example3:Solve the equation .
Let ,
Therefore
and
Example 4 : Solve the following simultaneous equations, giving your answers as exact fractions.
We aim for equations connecting x and y with no powers or logarithms.
Substituting,
C2 p33 Ex 3E
9. Changing the Base of a Logarithm
The log button on a calculator gives logarithms to base 10. However, with a little enterprise, we can still calculate logarithms to other bases.
Example 1 : Find .
Let
In general, to find , when we only know logarithms to base c...
And therefore…
A special case of this is when . Then
and so…
Example 2 : Solve the equation .
The logarithms are to different bases, so we change them to the lowest (and therefore simplest) base, 3. So the first logarithm needs changing from base 9. Let
Getting the base to be 3,
Substituting this back into the equation,
Example 3 : Solve the equation .
We change the second logarithm to base 5.
Let
Substituting in the equation,
Let ,
And so…
C2 p46 Ex 3F Topic Review : Exponentials and Logarithms