# A-level Mathematics/CIE/Pure Mathematics 2/Logarithmic and Exponential Functions
## Logarithms and Exponents

A logarithm is the inverse function of an exponent.
e.g. The inverse of the function $f(x)=3^{x}$ is $f^{-1}(x)=\log _{3}x$.
In general, $y=b^{x}\iff x=\log _{b}y$, given that $b>0$.

## Laws of Logarithms

The laws of logarithms can be derived from the laws of exponentiation:
${\begin{aligned}x^{a+b}=x^{a}\times x^{b}&\iff \log a+\log b=\log ab\\x^{a-b}=x^{a}\div x^{b}&\iff \log a-\log b=\log a/b\\(x^{a})^{b}=x^{ab}&\iff \log a^{b}=b\log a\end{aligned}}$
These laws apply to logarithms of any given base

## Natural Logarithms

The natural logarithm is a logarithm with base $e$, where $e$ is a constant such that the function $e^{x}$ is its own derivative.
The natural logarithm has a special symbol: $\ln x$
The graph $y=e^{kx}$ exhibits exponential growth when $k>0$ and exponential decay when $k<0$. The inverse graph is $y={\frac {1}{k}}\ln x$. Here is an interactive graph which shows the two functions as inverses of one another.

## Solving Logarithmic and Exponential Equations

An exponential equation is an equation in which one or more of the terms is an exponential function. e.g. $5^{x}=2^{x+2}$. Exponential equations can be solved with logarithms.
e.g. Solve $3^{x+1}=4^{2x-1}$
${\begin{aligned}3^{x+1}&=4^{2x-1}\\(x+1)\ln 3&=(2x-1)\ln 4\\x\ln 3+\ln 3&=2x\ln 4-\ln 4\\\ln 3+\ln 4&=x(2\ln 4-\ln 3)\\x&={\frac {\ln 3+\ln 4}{2\ln 4-\ln 3}}\\x&\approx 1.4844\end{aligned}}$
A logarithmic equation is an equation wherein one or more of the terms is a logarithm.
e.g. Solve $\lg x+\lg(x+2)=2$
${\begin{aligned}\lg x+\lg(x+2)&=2\\\lg(x(x+2))&=2\\x(x+2)&=100\\x^{2}+2x&=100\\(x+1)^{2}&=101\\x+1&={\sqrt {101}}\\x&=-1\pm {\sqrt {101}}\end{aligned}}$

## Converting Relationships to a Linear Form

In maths and science, it is easier to deal with linear relationships than non-linear relationships. Logarithms can be used to convert some non-linear relationships into linear relationships.

### Exponential Relationships

An exponential relationship is of the form $y=ab^{x}$. If we take the natural logarithm of both sides, we get $\ln y=\ln a+x\ln b$. We now have a linear relationship between $\ln y$ and $x$.
e.g. The following data is related with an exponential relationship. Determine this exponential relationship, then convert it to linear form.
| x | y |
| 0 | 5 |
| 2 | 45 |
| 4 | 405 |
${\begin{aligned}{\text{Exponential relationship }}\implies y&=ab^{x}\\5&=ab^{0}=a(1)\\a&=5\\y&=5b^{x}\\45&=5b^{2}\\9&=b^{2}\\b&=3\\y&=5(3^{x})\end{aligned}}$
Now convert it to linear form by taking the natural logarithm of both sides:
${\begin{aligned}y&=5(3^{x})\\\ln y&=\ln 5+x\ln 3\end{aligned}}$

### Power Relationships

A power relationship is of the form $y=ax^{b}$. If we take the natural logarithm of both sides, we get $\ln y=\ln a+b\ln x$. This is a linear relationship between $\ln y$ and $\ln x$.
e.g. The amount of time that a planet takes to travel around the sun (its orbital period) and its distance from the sun are related by a power law. Use the following data to deduce this power law:
| Planet | Distance from Sun /10$^6$ km | Orbital Period /days |
| Earth | 149.6 | 365.2 |
| Mars | 227.9 | 687.0 |
| Jupiter | 778.6 | 4331 |
${\begin{aligned}{\text{Power law}}\implies T&=aR^{b}\\{\text{Use Earth data}}\implies 365.2&=a(149.6^{b})\\\ln 365.2&=\ln a+b\ln 149.6\\{\text{Use Mars data}}\implies 687.0&=a(227.9^{b})\\\ln 687.0&=\ln a+b\ln 227.9\\\ln 687.0-\ln 365.2&=\ln a-\ln a+b\ln 227.9-b\ln 149.6\\\ln {\frac {687.0}{365.2}}&=0+b(\ln {\frac {227.9}{149.6}})\\b&={\frac {\ln {\tfrac {687.0}{365.2}}}{\ln {\tfrac {227.9}{149.6}}}}\approx 1.5011\\\ln 365.2&=\ln a+1.5011\ln 149.6\\\ln a&=\ln 365.2-\ln 1839.9\\\ln a&=\ln 0.1985\\a&=0.1985\\\therefore T&=0.1985R^{1.5011}\end{aligned}}$

#### References

* ↑ Retrieved from NASA's Planetary Fact Sheet

#### Notes

* ↑ $\lg$ is another way of writing $\log _{10}$
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