Exponents and Logarithms

Exponents

An exponential function is a function that involves repeated multiplications of something by itself. The most basic exponentional function can be expressed as $$a^n=b$$ where $a$ is any positive number that is being multiplied and $n$ is the number of times $a$ is multiplied by itself. $a$ is called the base, $n$ is called the exponent and $b$ is called the power. $$a^n= \! \underbrace{a\cdot a\cdot a \cdots a\,}_\text{$n$ times}$$

Properties of exponentials

$$\begin{align}x^a\cdot x^b&=x^{(a+b)}\\ \frac{x^a}{x^b}&=x^{(a-b)}\\(x^a)^b&=x^{(a\cdot b)}\\a^{(-n)}&=\frac{1}{a^n}\\x^0&=1\\x^{\frac{a}{b}}&=\sqrt[b]{x^a}\end{align}\\\text{if }a^x=a^y\text{, then: }x=y$$

Example

Take $$2^4$$ This exponential can be re-written as $$2\cdot 2\cdot 2\cdot 2 = 4\cdot 2\cdot 2=8\cdot 2=16$$&

Thus:$$2^4=16$$

Logarithms

Logarithms are the "opposites" of exponentials. They are used to undo exponential functions. Logarithmic expressions are written in the form: $$\log_{a}b=c$$ where $a$ is what is called the base of the logarithm and $b$ is the antilogarithm and $c$ is the logarithm.

Properties of Logarithms

$$\begin{align}\log_{b}a+\log_{b}c&=\log_{b}{(a\cdot c)}\\\log_{b}a-\log_{b}c&=\log_{b}\left(\frac{a}{c}\right)\\\log_{b}{(a^c)}&=c\cdot log_{b}a\end{align}$$

Logarithms and Exponents Practice Problems

Solve $$\log_{x}49=2$$

By the property of logarithms and exponents, we can rewrite the equation as: $$49=x^2$$And so$$\sqrt{49}=x$$$$7=x$$