Notation 1
$${a^x = \underbrace{a \cdot a \cdot a \cdot a \cdot … \cdot a}_\text{x number of factors of a} }$$
where a is called the base (Basis) and x is the exponent or power (Potenz bzw Hochzahl)
Rules of powers 2
If the base is the same.
$${a^m \cdot a^n = a^{m+n}}$$
$${a^m : a^n = \frac{a^m}{a^n} =a^{m-n}}$$
$${(a^m)^n = a^{m \cdot n}}$$
$${a^1 = a}$$
$${a^0 = a^{m-m} = \underbrace{\frac{a \cdot a \cdot a \cdot a \cdot … \cdot a}{a \cdot a \cdot a \cdot a \cdot … \cdot a}}_\text{m number of factors a} = \frac{1}{1}= 1}$$
If the base is different but exponent is the same
$${a^n \cdot b^n = (ab)^{n}}$$
$${\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^{n}}$$
Negative powers 3
$${\frac{1}{a} = \frac{1 \cdot a}{a \cdot a} = \frac{a^1}{a^2} =a^{1-2} = a^{-1}}$$
units in physics are sometimes written using negative powers.
velocity in meters per second $${\text{velocity v in }~\frac{m}{s}= ms^{-1}}$$
acceleration in meters per second squared $${\text{acceleration a in }~\frac{m}{s^2}= ms^{-2}}$$
Standard notation - Zehnerpotenzschreibweise 4
Standard notation or scientific notation means writing a number with one digit before the decimal sign multiplied by a power to the base ten.
$${53716 = 5.3716 \cdot 10000 = 5.3716 \cdot 10^4}$$
$${0.00028014 = 2.8014 \cdot 0.0001 = 2.8014 \cdot 10^{-4}}$$
In general,
$${k \cdot 10^z~,~ z \in \mathbb{Z}~ \text{and}~ 1 \leq k < 10}$$
Rational exponents 5
$${(a^2)^{\frac{1}{2}} = a^{2 \cdot \frac{1}{2}} = a^1 = a}$$
But,
$${\sqrt{(a^2)} = a}$$
So, taking a number to the power of one half is the same as taking the square root.
$${b^{\frac{1}{2}} = \sqrt{b}}$$
$${b^{\frac{1}{3}} = \sqrt[3]{b}}$$
In general,
$${a^{\frac{1}{n}} = \sqrt[n]{a}}$$
$${a \geq 0 ~\text{for n even and } ~ ~ a \in \mathbb{R} ~\text{for n odd} }$$
exponential equation - exponentielle Gleichung
An equation with the variable in the exponent is called an exponential equation.
example
$${16 = 2^x}$$ $${2^4 = 2^x}$$ $${4 = x}$$
example
$${2^{x + 3 } = 2^{1-x}}$$ $${x + 3 = 1 - x}$$ $${2x = -2}$$ $${x = -1}$$
If you manage to write both sides as a power of the same base, it may be possible to solve the expressions without the use of logarithms.
Even ones like this:
$${0.25^{2-x} = \frac{256}{2^{x +3 }}}$$
exponential function - Potenzfunktion
The basic exponential function is defined as:
$${f(x) = a^x}$$
example
$${f(x) = 2^x}$$
Stretching the function by a factor k you can write.
$${f(x) = k \cdot a^x}$$