(last updated: 2:12:14 PM, October 02, 2020)
Basic exponents:
\[b^2 \quad \text{ means } \quad b\cdot b\] \[b^3 \quad \text{ means } \quad b\cdot b\cdot b\] \[b^{3/4} \quad \text{ means } \quad \sqrt[4]{b^3}\]
The number \(b\) is called the base and we usually require \(b>0\). It is ok if \(b<0\) when raising to an odd power or odd root. \(b=0\) is ok with positive powers.
Some more advanced properties:
\[b^0=1\] \[b^1=b\] \[(b^x)^y=b^{xy}\] \[b^{n/m}=\sqrt[m]{b^n}=(\sqrt[m]{b})^n\] \[b^xb^y=b^{x+y}\] \[\frac{b^x}{b^y}=b^{x-y}\] \[b^{-x}=\frac1{b^x}\]
\(b^x\) is called an exponential expression or just an exponential for short. We say that we are “exponentiating \(b\) with \(x\)” or that we are “raising \(b\) to the \(x\) power” or simply “\(b\) to the \(x\).”
Examples:
\[2^3=8\] \[125^{2/3}=(\sqrt[3]{125})^2=(5)^2=25\] \[10^{2.73}\approx 537.03179637 \quad \text{ (via calculator)}\]
Scientific notation takes the form \(x.xxxx(10)^y\) where the number of decimal places included is a matter of preference or significant figures.
Examples:
\[2.7(10)^6=2700000 = 2,700,000\quad \text{(this is 2.7 million)}\] \[3.051(10)^{-4}=0.0003051\] Calculators often give it written as: \[7.23E-9\] but this just means \(7.23(10)^{-9}\).
\[\log_b(M)=x \quad \text{ means that } \quad b^x=M\]
\[\log_b(1)=0\] \[\log_b(b)=1\] \[\log_b(b^x)=x\] \[b^{\log_b(M)}=M\] \[\log_b(N\cdot M)=\log_b(N)+\log_b(M)\] \[\log_b\left(\frac{N}{ M}\right)=\log_b(N)-\log_b(M)\] \[\log_b(M^r)=r\log_b(M)\]
Also “\(\log\)” means \(\log_{10}\)" (log base 10) and “\(\ln\)” means “\(\log_e\)” (natural log). THe number \(e\) is called the natural exponential or Euler’s number, \(e\approx 2.718\).
Examples:
\(\log_28=3\) because \(2^3=8\)
\(\log(100)=2\) because \(10^2=100\)
\(\ln(e^5)=5\)
Here is a tricky one:
\(\log_8(16)=?\) well, let \(\log_8(16)=x\) and that tells us that \(8^x=16\) but these are both powers of \(2\) so we can say \((2^3)^x=(2^4)\) i.e. that \(2^{3x}=2^4\). So \(3x=4\) thus \(x=\frac43\).
\[ \diamond \S \diamond \]