A-level Mathematics/OCR/C2/Logarithms and Exponentials

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Operations With Exponential Function[edit]

An exponential function is a function where a constant base (b) is raised to a variable.

Multiplication[edit]

Firstly, b^x \times b^{2x} is b^{\left(x + 2x\right)}\, which is b^{3x}\,. So when you multiply a base by the same base you add the variables. To clarify, here is an example with numbers:

x\, 2^x\, 2^{2x}\, 2^x \times 2^{2x}\, 2^{3x}\,
1 2 4 8 8
2 4 16 64 64

Division[edit]

Secondly \frac{b^{2x}}{b^y} is b^{\left( 2x - y \right)}\,(also y \neq 0). So when a base is divided by the same base you subtract the variables.

Here is an example with numbers: \frac{2^4}{2^2}=\frac{16}{4}=4=2^2.

Base raised to two powers[edit]

Thirdly \left(b^{2x}\right)^{3x} is b^{\left(2x\right) \times \left(3x\right)} which is b^{6x^2}. So when a base with a variable is raised to a variable you multiply the variables. Here is another example with numbers: (when x = 1) \left(2^2\right)^3=4^3=64=2^6.

Multiple bases[edit]

Fourthly when a^2 \times b^2 = ab \times ab it is the same as \left(ab\right)^2\,. Here is an example with numbers: 2^2 \times 3^2 = 36 = 6^2. There is a similar situation with division: \left(\frac{a}{b}\right)^2 = \frac{a}{b} \times \frac{a}{b} = \frac{a^2}{b^2}. So when you multiple or divide two different bases raised to the same variable you can multply or divide them first and then raise them to the variable.

Fractional exponents[edit]

The last case is when x is presented as a fraction, you can make a square root function, for example b^\frac{1}{x} becomes \pm \sqrt[x] b. However it is customary to only use the positive root and so b^\frac{1}{x} is defined as \sqrt[x] b. Another similar case is when the fraction has a constant (designated as c) other than 1 in the numerator , for example b^ \frac {3}{x} = \left( \sqrt[x] b \right)^3 so b^ \frac {c}{x} = \left( \sqrt[x] b \right)^c.

The Laws of Exponents[edit]

The rules that have been suggested above are known as the laws of exponents and can be written as:

  1. b^xb^y = b^{x+y}\,
  2. \frac{b^x}{b^y} = b^{x-y}
  3. \left(b^x\right)^y = b^{xy}
  4. a^n b^n = \left(ab\right)^n\,
  5. \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
  6. b^{-n}=\frac{1}{b^n}
  7. b^ \frac {c}{x} = \left( \sqrt[x] b \right)^c where c is a constant
  8. b^1=b\,
  9. b^0=1\,

Solving exponential equations[edit]

In order to solve an exponential equation you need to make sure that all the bases are the same. Then you can remove the base and solve for the variable. Here is an example:

Solve for x. 2^{\left(x-1\right)} = 16\,

Now we convert 16 to a base 2 raised to a number.

2^{\left(x-1\right)} = 2^4\,

Now we can remove the base. So we have:

x-1 = 4\,

Finally solve for x.

x = 5\,

Graphing an Exponential Function[edit]

When you graph an exponential function you use the same methods as with a regular function. There is a graph below that you can look at.

Logarithmic Functions[edit]

Logexponential.svg

In mathematics you can find the inverse of an exponential function by switching x and y around: y = b^x\, becomes x = b^y\,. The problem arises on how to find the value of y. The logarithmic function solved this problem. All conversions of logarithmic function into an exponential function follow the same pattern: x = b^y\, becomes y = \log_b x\,. If a log is given without a written b then b=10. Also with logarithmic functions, b > 0 and b \ne 1. There are 2 cases where the log is equal to x: \log_bb^X = X\, and b^{\log_bX} = X\,.

Laws of Logarithmic Functions[edit]

When X and Y are positive.

  • \log_bXY = \log_bX + \log_bY\,
  • \log_b \frac{X}{Y} = \log_bX - \log_bY\,
  • \log_b X^k = k \log_bX\,

Change of Base[edit]

When x and b are positive real numbers and are not equal to 1. Then you can write \log_a x\, as \frac { \log_b x}{ \log_b a}. This works for the natural log as well. here is an example:

\log_2 8 = \frac { \log 8}{ \log 2} = \frac {.9}{.3} = 3\, now check 2^3 = 8\,

This is part of the C2 (Core Mathematics 2) module of the A-level Mathematics text.


Dividing and Factoring Polynomials / Sequences and Series / Logarithms and Exponentials / Circles and Angles / Integration

Appendix A: Formulae