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Logarithms

Logarithms with base $b$

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.

Log and exponent graph

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 Rules

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: $(a^m)^n = a^{mn}$

$x=1/3$

Logarithms with base $10$

$10^{-3}=0.001$, 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$_{5}25 = {log_{10}25}/{log_{10}5} = {1.398}/{0.699} = 2$

Exponential Equations using Logs

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$

Growth and Decay

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.

Exponential growth
Exponential growth: rabbits and their habits

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).

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