> See also: > - # Logarithmic and Exponential Functions ## Exponential Functions > **Preface:** > It's important to refresh yourself on the rules of exponential operations before tackling more complex fields of mathematics. > > It can be simple to handle the other operations at higher levels of math, but if you forget the exponential rules you will always find yourself lost. In the expression $a^b$, $a$ is known as the **base** and $b$ is known as the **exponent**. **Rational Exponents:** aaa ![[Pasted image 20220515142230.png|400]] **Radical Expressions:** The form of exponents which uses the radical symbol ($\sqrt{a}$). If no number is present above the radical it is known as a "square root", however, if one is present it is known as the "nth root" depending on the value. - Ex: $\ --- The expression $t^{\frac{3}{4}}$ is the same as $t^{\frac{1}{4}} \cdot t^{\frac{1}{4}} \cdot t^{\frac{1}{4}}$ ## Properties of Exponents ### Exponential Rules The systems described above can be simplified into a set of **exponential rules** which describe what operations can be performed on both rational exponents and radical expressions. > [!info]+ Product Rule ![[Pasted image 20220515141818.png]] Difference of Squares: $a^2-b^2=(a+b)(a-b)$ --- Difference of Cubes: $a$ ## Logarithmic Functions When you raise a number to Because logarithms are the inverse of exponents, when you take the logarithm of a number you are essentially Logarithms and exponentials with the same base are inverse functions. $f(x)=b^2$ vs $a=3$ $log_a(b) = b^a$ $log_a(x) + log_a(y) = log_a(x+y)$ $log_a(x) - log_a(y) = log_a(x-y)$ $log_e=ln$ $log_{10}=log$ ### Natural Logarithms $ln(e^x)=x=e^{ln(x)}$ ![[Pasted image 20240410034128.png|400]] #### The Natural Logarithm as an Integral ## Properties of Logarithms If $a,b,c>0$, $b\neq a$, and $r$ is any real number, then the following properties are true: > [!rule]+ Logarithmic Properties > Power Rule: > $log_b(a^c)=c \cdot log_b(a)$ > --- > **Product Rule:** > $log_b(ac)=log_b(a)+log_b(c)$ > --- > **Quotient Rule:** > $log_b(\frac{a}{c})=log_b(a)-log_b(c)$ > --- > **Change of Base Rule:** > $log_b a = \frac{log_n a}{log_n b}$ > --- > **Equality Rule:** > $log_b a = log_b c \iff a=c$