Indices, logs and exponentials

YouTube lecture recording from October 2020

Indices

(1) Multiplication of expressions with powers

$5^2 = 5 \times 5 = 25$,

$5^3 = 5 \times 5 \times 5 = 125$.

$5^5 = 5 \times 5 \times 5 \times 5 \times 5 = 3125$

$25 \times 125 = 3125$.

$5^5 = 5^2 \times 5^3$.

$a^m\times~a^n=a^{m+n}$.

(2) Division of expressions with powers

$\frac{5^3}{5^2} = \frac{5 \times 5 \times 5}{5 \times 5}$

${a^n\over a^m}=a^{n-m}$

(3) Powers of expressions with powers

$\left(5^3\right)^4 = 5^3 \times 5^3 \times 5^3 \times 5^3 = 244140625$

$5^{12} = 244140625$.

$(a^n)^m=a^{nm}$

(4) Negative powers

$\frac{a^3}{a^5} = \frac{a \times a \times a}{a \times a \times a \times a \times a} = \frac{1}{a \times a} = \frac{1}{a^2}$

$a^{-2}={1\over a^2}$

$a^{-n} ={1\over a^n}$

(5) Zero index

$\frac{a^3}{a^3} = \frac{a \times a \times a}{a \times a \times a} = 1$,

$a^0=1$

(6) Fractional (rational) indices

$a^1 = a^{1/2} \times a^{1/2} {\rm ~~~~so~~~~}a^{1/2} = \sqrt{a}$ $a^{1\over n} = \sqrt[n]{a}$ $a^{n\over m} = \left(\sqrt[m]{a}\right)^n ~~~~~{\rm or}~~~\sqrt[m]{a^n}$ $a^{-{n\over m}} = {1\over {\left(\sqrt[m]{a}\right)^n}}~~~{\rm or}~~~~{1\over {\sqrt[m]{a^n}}}$

(7) Different bases, same index

$(a\cdot b)^x = a^x\cdot b^x\qquad {\rm and}\qquad\Bigl( {a\over b}\Bigr)^x={{\,a^x}\over{\,b^x}}$

(8) Two Cautionary remarks

1. Powers of sums are not pretty: $(a+b)^x \ne a^x + b^x$
2. Powers of differences are not pretty: $(a-b)^x \ne a^x - b^x$

1. $(a+b)^2=(a+b)(a+b)=a^2+2ba+b^2$
2. $(a-b)^2=(a-b)(a-b)=a^2-2ba+b^2$
3. $(a+b)(a-b)=a^2-b^2$

Logarithms

1. If $y=a^x$, $\;x\;$ is the power to which the base $\;a\;$ must be raised to give $y$, OR
2. $\;x\;$ is the logarithm which, with base $\;a,\;$ gives $\;y\;$.

Getting a feel for logarithms

• volume of sound (decibels)
• pitch (1 octave is doubling of frequency, 2 octaves is quadrupling of frequency)
• earthquakes (Richter scale)
• pH ($[H^+]$ ion concentration)
• Used a lot in computer programming to improve the speed and/or accuracy of calculations.

The laws of logarithms

(1) Using the same base $a$ for both operations

• Taking the logarithm undoes raising to a power:

$\log_a\,a^r=r$

• Raising to a power undoes taking the logarithm:

$a^{\log_a\,b}=b$

(2) Multiplication

$\log_a (bc) = \log_a b + \log_a c ~~~~~(Add)$

(3) Division. In a similar way to multiplication

$\log_a \left({b \over c}\right) = \log_a b - \log_a c ~~~~~(Subtract)$

(4) Powers

$\log_a b^n = n \log_a b ~~~~~~ (Multiply)$

(5) Changing the base of a logarithm

$\log_a c = {\log_b c\over \log_b a}$

(6) Special case: if $b$ and $c$ are the same, (5) reduces to

$\log_a b ={\log_b b\over \log_b a}={1\over \log_b a}$

(7) The log of any number to the base itself is 1

$\log_a a =1$

(8) The log of 1 is 0 (unless a=0, in which case it is undefined)

$\log_a 1 = 0 \quad{\rm since~~~}\quad a^0=1$

(9) Inverse operation

$\log_a a^x = x$

(10) Or

$a^{\log_a x}=x$

(11) Negative logs

$\log_a {1\over x} = \log_a1-\log_ax= 0 - \log_a x =-\log_ax$

(12) Two cautionary remarks

1. $\;\log_a (x + y)\;$ and $\;\log_a (x-y)\;$ cannot be simplified any further, and should be left as they are.
2. Neither can $\;\log_a\,x \cdot \log_a\,y\;$ or $\;\displaystyle{{\log_a\,x}\over {\log_a\,y}}.\;$ Leave them as they are.

Example 1: graphs

$\log y =\log Ax^n = \log A + \log x^n= \log A + n \log x$

$y=3.0\times x^{1/2}$

Example 2: pH

1. What is the pH of a 0.011M solution of HCl?

   $pH = -\log_{10}[H^+]$

   Copy
   import numpy as np
   pH = -np.log10(0.011)
   print('pH =',pH)

   Copy
   pH = 1.958607314841775

2. What is the H$^+$ concentration of a solution of HCl with a pH of 3?

   $pH = -\log [H^+] = 3~~~~{\rm~so~~~}$

   Copy
   [H+] = 0.001 M

Example 3: Simplifying logs

$\log(x) = \log(p) + 2 \log(q) - \log(k) -3$ > $\log(x) =~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ > $x=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

1. Use the laws of logarithms (above) to put all terms on the right hand side within the logarithm. This means we have to rewrite $\;3\;$ as $\;3\log(10)\;$.
2. Raise both sides to the (same) power which will undo the logarithm function. (i.e. raise both sides to the power of 10)

The exponential function and the real number e

Definition of the Real Number $e$

$y=e^x ~~~{\rm the~gradient~at~(0,1)~is~~}1 = e^0.$

$y=\log_e10=\ln 10 {\rm~~and~~~} y=\log_ex=\ln x$

Definition of the natural logarithm

• $\ln(e)=\log_e(e)=1$
• $\ln(10)=\log_e(10)$ = "The power I need to raise $e$ to in order to get 10"
• $\ln(10)\approx 2.303$. Which means that $e^{2.303}\approx 10$
• $\ln(\exp(b)) = \ln e^b =\log_e e^b = b$
• $\exp(\ln(b))=e^{\ln(b)}=e^{\log_e b}=b$

Introductory problems

Main problems