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  • Revision:OCR Core 2 - Exponentials and logarithms

TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > OCR Core 2 - Exponentials and logarithms


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7. Exponentials and logarithms

Exponential functions

A simple way to define an exponential function is simple a function involving a constant to the power of a function of a variable. The reason for the name is that the word exponent refers to a power.

For example:

 y = 2^{x}

The values which are associated with the graph of this function are a geometric sequence, but when this is graphed one can refer to it as an exponential function.

The graph of the aforementioned function looks like a solid line, however it does have undefined values. Irrational values of x do not give a meaning to the function, and only in Core 3 will a method be developed such that one can define these.

The only restrictions on the base of an exponential function is that it must be positive, real, and not 1.

A general form for the exponential function is:

 f(x) = b^{x}

The letter "b" is conventionally used due to it being the base of the function.

Consider the case that "b" was below zero. One is already aware that one can express the square root, and other roots as fractions ( \sqrt[n]{x} = x^{ \frac{1}{n}} ), and one should be aware that there is no real number such that when it is squared, it is negative. Hence one can deduce that if "b" were negative, there would be gaps, as the function cannot be defined for many values (including  x = \frac{1}{2} ).

If "b" was 0, one would have an undefined function for any negative value of x, as  b^{-x} = \frac{1}{b^{x}} , and if  b = 0 ,  b^{x} = 0 , and division by zero is not defined.

One is excluded for the same reason that it is excluded from being the common ratio in geometric sequences, it does not produce a useful function, merely a constant value (for any value of x).

One might notice that if one plots graphs of:

 y = b^{x} ,

and:

 y = \left( \frac{1}{b} \right) ^{x} ,

the latter is a reflection in the y-axis of the former.

This is simple to explain:

 y = \left( \frac{1}{b} \right) ^{x} = \frac{1^{x}}{b^{x}} = b^{-x}

One knows from the transformation of graphs that:

 f(x) \to f(-x)

results in a reflection in the y-axis.

This is the same idea, as one is going from:

 b^{x} \to b^{ -x} .

All of the graphs of exponential functions go through the point (0, 1), this is because for any "b" (for which the function is exponential),  b^{0} = 1 .

One should also note that as it is impossible to obtain a negative number from one of these exponential functions, the graphs are above the x-axis.

Due to the reflection in the y-axis, one can explain the shapes of both "types" of graph (one type being where  0 < b < 1 , and the other where  b > 1 ).

Consider  b > 1 .

As "x" is negative, the value will decrease more and more, and as "x" becomes positive the value will increase more and more. This gives the shape of a graph which slopes upwards from the left side of the y-axis, to infinity.

The other "type" of the exponential function is merely the opposite. These functions have a negative gradient, and are highest fore negative "x".

Logarithms

One can observe from the graphs which one has seen of the exponential functions, that there is a single value of x for which  b^{x} is a given value.

A logarithm is the inverse function of an exponential function, such that:

If:

 y = b^{x}

 \log _{b} y = x

The concept might require some thinking, however it is simply understood that the logarithm to a base produces the value to which the base must be raised such that the output is the number one is taking the logarithm of.

For example:

 \log _{10} 100 = 2 .

One should be aware of several special cases.

 \log _{b} b = 1

(As  b = b^{1} ).


 \log _{b} 1 = 0

(As  1 = b^{0} ).

One can show two identities:

 \log _{b} b^{x} \equiv x

One can easily see that this is correct, as one will have to raise the base to the power "x" in order to obtain  b^{x} .

Also:

 \displaystyle b^{ \log _{b} x } \equiv x

This is simply a rearrangement, and shows how the logarithm "cancels out" the raising to the power.

Properties of logarithms

The rules for logarithms are the opposite to those for powers (as one might expect).

Hence:

 \log _{b} x + \log _{b} y = \log _{b} (x \times y) .

 p = \log _{b} x

 q = \log_{b} y

Hence:

 b^{p} = x

 b^{q} = y

Therefore:

 x \times y = b^{p} \times b^{q} = b^{p + q}

Now consider the logarithm of this:

 \log _{b} \left( b^{p + q} \right) = p + q = \log _{b} x + \log _{b} y .

This is called the multiplication rule.

Now one can consider the division rule, using the same variables as before:

 \displaystyle \frac{x}{y} = \frac{b^{p}}{b^{q}} = b^{p - q}

 \displaystyle \log _{b} \left( b^{p - q} \right) = p - q = \log _{b} x - \log _{b} y

Another rule is the power rule:

 x^{n} = \left( b^{p} \right) ^{n} = b^{pn}

 \displaystyle \log _{b} \left( b^{pn} \right) = pn = n \log _{b} x .

Another rule is basically an application of the power rule:

 \displaystyle \log_{b} \left( b^{pn} \right) = pn = n \log _{b} x

Hence, as:

 \displaystyle \sqrt[m]{n} = n^{ \frac{1}{m} } ,

 \log _{b} \left( \sqrt[m]{n} \right) = \log _{b} \left( n^{ \frac{1}{m} } \right) = \frac{1}{m} \times \log _{b} n = \frac{ \log _{b} n}{m} .

Special bases

Generally there are two bases that are in use, e, and 10.

One would use a logarithm to the base 10 for a logarithmic scale. The logarithmic scale is a scale in which a set distance represents exponentially increasing intervals.

Equations and inequalities

One can use the different rules for the logarithms in order to solve equations.

Consider the following:

 2^{x} = 4

One might be able to immediately spot that x = 2.

One can use logarithms however:

 \log \left( 2^{x} \right) = \log 4 = x \log 2

These logarithms can be to any base.

 x = \frac{ \log 4 }{ \log 2 } = 2

If one is dealing with inequalities, one must be careful as some logarithms are negative, and therefore division by them will cause a reverse of the inequality.

One question which one might address is whether larger numbers always have larger logarithms.

One must consider two cases, where the base is  0 < b < 1 , and when the base is  b > 1 .

Now consider the graphs of the exponential functions which were discussed previously. If the x-coordinate is  \log _{b} y , the y-coordinate is evidently  y .

For  b > 1 due to the ever-increasing function, the greater y becomes, the greater the logarithm.

With  0 < b < 1 , the function is ever-decreasing, and therefore as the value of y increases, the logarithm will decrease.

This means that one must be careful, if one is dealing with a base that is below 1, the inequality is reversed when one takes the logarithm. It is rare, in practice, for a logarithm of this type to be used, and therefore one does not generally have to deal with this issue.


Also See

Read these other OCR Core 2 notes:

  1. Trigonometry
  2. Sequences
  3. The binomial theorem
  4. The sine and cosine rules
  5. Integration
  6. Geometric sequences
  7. Exponentials and logarithms
  8. Factors and remainders
  9. Radians
  10. The trapezium rule
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