Question

What are the Exponential and Logarithmic Properties/Laws?

Answer 1

Exponential Properties:
1. Product of like bases: To multiply powers with the same base, add the exponents and keep the common base
$$
a^{m} a^{n}=a^{m+n}
$$
2. Quotient of like bases: To divide powers with the same base, subtract the exponents and keep the common base
$$
\frac{a^{m}}{a^{n}}=a^{m-n}
$$
3. Power to a power: To raise a power to a power, keep the base and multiply the exponents.
$$
\left(a^{m}\right)^{n}=a^{m n}
$$
4. Product to a power: To raise a product to a power, raise each factor to the power. $(a b)^{m}=a^{m} b^{m}$
5. Quotient to a power: To raise a quotient to a power, raise the numerator and the denominator
to the power. $\left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}}$
6. Zero exponent: Any number raised to the zero power is equal to "1".
$$
a^{0}=1
$$
7. Negative exponent: Negative exponents indicate reciprocation, with the exponent of the reciprocal becoming positive. You may want to think of it this way: unhappy (negative) exponents will become happy (positive) by having the base/exponent pair "switch floors"!
$$
a^{-n}=\frac{1}{a^{n}} \quad \text { or } \quad \frac{1}{a^{-n}}=a^{n}
$$

Definition of the Logarithmic Function:

$\log _{a} x=y \quad \leftrightarrow \quad a^{y}=x \quad$ The word "log" asks: What power do I put on 2 to get 8 ?

$\log _{2} 8=3 \quad \leftrightarrow \quad 2^{3}=8 \quad$ Answer: 3

 

Common Logarithm:

The logarithm with base 10 is called the Common Logarithm and is denoted by omitting the base.
Properties:
1. $\log _{a}(A B)=\log _{a} A+\log _{a} B$
2. $\log _{a}\left(\frac{A}{B}\right)=\log _{a} A-\log _{a} B$
3. $\log _{a}\left(A^{c}\right)=C \cdot \log _{a} A$

 

Natural Logarithm:
The logarithm with base $e$ is called the Natural Logarithm and is denoted by 'In'.
Change of Base:
This formula allows you to find the calculator value of the log of any base.
$$
\log _{b} x=\frac{\log _{a} x}{\log _{a} b}
$$