Logarithms are a useful way of recording and making calculations. The log is the opposite of exponential expressions:

If $y = b^x$, then $x = log_by$

where $b$ is the base, and $x$ is the exponent.

The domain (permitted x values) of the logarithm function $y = b^x$ is x > 0 (set of positive real numbers), and the range (resulting y values) is all real numbers (y ∈ ℜ).

log$xy$ = log$x$ + log$y$

log$x/y$ = log$x$ - log$y$

log$x^n = n$log$x$

log$_a{1/x}$ = log$_ax^{-1}$ = -log$_ax$

log$_ba $ = ${log_ca}/{log_cb}$

log$_a(a^x) = x$

$a^{log_ax} = x$

Examples

1. Evaluate $log_4{64}$ $x = log_4{64}$ $x = log_4{4^3}$ $x = 3log_4{4}$ //rule: $log_ab^c = c⋅log_ab$ $x=3$ //rule: $log_aa = 1$ |
2. Evaluate $log_{125}5$ $x = log_{125}5$ $125^x = 5$ //rule: $a^{log_ax} = x$ $(5^3)^x = 5^1$ $3x=1$ //rule: $(a^m)^n = a^{mn}$ $x=1/3$ |

$10^{-3}=0.001$, so $log0.001=-3$

$10^{-2}=0.01$, so $log0.01=-2$

$10^{-1}=0.1$, so $log0.1=-1$

$10^{0}=1.0$, so $log1.0=0$

$10^{1}=10$, so $log10=1$

$10^{2}=100$, so $log100=2$

$10^{3}=1000$, so $log1000=3$

$10^{1/2}=√{10}$, so $log√{10}=1/2$

Since the most common number system is decimal, or base 10, if a log does not specify a base, it is assumed to be ten.

$a=10^{loga}$, where $a>0$. $a$ must be greater than zero because there is no power to which any number can be taken which produces zero or a negative number.

log$ab$ = log$a$ + log$b$: to test this rule, let us find the log of 20.

log$_{10}20$ = log$_{10}2$ + log$_{10}10$ = log$_{10}2 + 1 = 1.301$

Recall that log$_ba = {log_ca}/{log_cb}$:

log$_{5}25 = {log_{10}25}/{log_{10}5} = {1.398}/{0.699} = 2$

Being able to manipulate logs and exponentials is very useful for solving certain types of equations.

For example, what is $x$ in $2^x=20$?

To solve, we take the log of both sides:

log$2^x=$log$20$ ⇒ $x$log$2=$log$20$ ⇒ $x={log20}/{log2}= {log2}/{log2}+ {log10}/{log2}= 1 + 1/{log2} = 1.0 + 3.32 = 4.32$

Examples

1. Show that log$(1/{16}) = -2$log$4$ log$(1/{16}) = $log$4^{-2}$ $= -2$log$4$ Apply rule: log$(a^b) = b$log$a$ |
2. Show that log$(250) = 3 - $log$4$ log$(250) = $log$({1000}/4)$ $= $log$(10^3) + $log$(4^{-1})$ $= 3 - $log$4$ Or apply rule: log$(a/b) = $log$a - $log$b$ |

The discovery of what is now called *Euler's Number (e)* was actually made by Jacob Bernoulli, another brilliant Swiss mathematician. He made the discovery through studying compound interest rates (typical Swiss bankers!...). *e* is the limit of compounding interest rates with ever shorter time intervals. 100 Swiss francs invested at 100% interest (highly theoretical!) and compounded continuously (i.e. the interest accrued is added continuously to the capital) would yield 271.82 Swiss francs at the end of the year.

This can be calculated from: $Y = C(1+r/n)^{nt}$, where Y is the yield (amount at the end of the year), C is the original capital (100 Fr.), $r$ is the interest rate in decimals (1.0 in our unrealistic example), $n$ is the number of times te interest is compounded (a value of 365 would mean the interested is calculated and added to the capital once a day), and $t$ is the number of years the investment is made for.

Many natural phenomena have exponential growth, such as populations.

The opposite of exponential growth is decay: examples are radioactive decay, and temperature change.

Population Growth

The population of rabbits grows according to the following exponential formula:

$Q(t) = 1+2^{0.15t}$

where $t$ is the number of weeks since the population was size $Q_0$ (2).

Content © Renewable.Media. All rights reserved. Created : November 7, 2014

The most recent article is:

View this item in the topic:

and many more articles in the subject:

Environmental Science is the most important of all sciences. As the world enters a phase of climate change, unprecedented biodiversity loss, pollution and human population growth, the management of our environment is vital for our futures. Learn about Environmental Science on ScienceLibrary.info.

1936 -

Edward Stone's name is synonymous with space exploration. For more than 50 years he has been a leading scientist behind the NASA unmanned probes to Mars, the giant planets, and the outer reaches of the solar system.

Website © renewable.media | Designed by: Andrew Bone