2.4.1 Definition & Notation
Logarithms are inverse functions of exponents
$$b^p=x \therefore \log_b x=p$$
Example
$$2^3=8 \therefore \log_2 8=3$$
2.4.2 Base Inverse Property
$$\log_b b^p=p$$
Proof
$$\log_b x = p \land x=b^p \therefore \log_b b^p=p$$
Example
$$\log_2 8=3 \land 8=2^3 \therefore log_2 2^3=3$$
2.4.3 Antilogarithms of an Equation
Given an equality, an equation may be raised as exponential powers with the same base value
$$a=b↔x^a=x^b$$
Proof
Given $x^a=x^b$, take the logarithm of base $x$
$$\log_x x^a=\log_x x^b$$
Apply the
base inverse property on each side
2.4.4 Power Inverse Property
$$ b^{\log_b x^p}=x^p, \forall x \isin \R$$
Deductive Logic
Logarithms are the inverse functions of exponents, therefore operating a logarithm in an exponent with the same base cancels both
Example
$$\log_2 2^3=2^{log_2 3}=3$$
2.4.5 Power Rule
$$\log_b x^p=p·\log_b x$$
Proof
Let $a=\log_bx$ so that $x=b^a$, then exponentiate to $p$
$$x^p={(b^a)}^p$$
Distribute the power
$$x^p=b^{a·p}$$
Take the logarithm using base $b$
$$\log_b x^p=\log_b b^{a·p}$$
Apply the
base inverse property
$$\log_b x^p=a·p$$
Substitute $a$ for its original form
Example 1
$$\log_2 64=\log_2 2^6=6·\log_2 2=6$$
Example 2
$$\log_2 x^{-1}=-\log_2 x$$
2.4.6 Product & Quotient Rules
$$\log_b (M·N^{\pm 1})=\log_b M \pm \log_b N$$
Proof
Let
$$ x=\log_b M \land ±y=\log_b N^{±1}$$
So that
$$M=b^x \land N^{±1}=b^{±y}$$
Multiply the two equalities
$$M·N^{±1}=b^x·b^{±y}$$
Apply the
exponent power rule
$$M·N^{±1}=b^{x±y}$$
Take the logarithm of base $b$
$$\log_b (M·N^{±1})=\log_b b^{x±y}$$
Apply the
base inverse property
$$\log_b (M·N^{±1})=x±y$$
Substitute $x$ and $±y$ for their original terms
2.4.7 Sum & Difference Rules
$$\log_b (M \pm N)=\log_b M+\log_b \Big(1 \pm \frac{N}{M} \Big)$$
Proof
Given $log_b(M±N)$, multiply $N$ by $M/M$
$$\log_b \Big(M \pm \frac{M}{M}·N \Big)$$
Factor
$$\log_b \Big(M· \Big( 1 \pm \frac{N}{M} \Big) \Big)$$
Apply the
product rule
2.4.8 Change of Base Rule
$$\log_b x = \frac{\log_a x}{\log_a b},$$
$$x>0\medspace\land\medspace a>0\medspace\land\medspace a≠1$$
Properties from the Change of Base Rule
$$1)\qquad \log_b x = \frac{1}{log_x b}$$
$$2)\quad\thinspace \log_{c^n} x=\frac{\log_c x}{n}$$
$$3)\qquad a^{\log_b x}=x^{\log_b a}$$
Proof
Let $c=\log_bx$ so that $b^c=x$, then take the logarithm of base $a$
$$\log_a b^c=\log_a x$$
Apply the
power rule
$$c·\log_a b=\log_a x$$
Solve for $c$ and sutstitute it for its original form
Proof of Properties
1) Using the change of base rule, substitute $a$ for $x$
2) Using the change of base rule, substitute $c^n$ for $b$ and $c$ for $a$
3) Take the logarithm $b$ of the given formula and apply the
power rule
2.4.9 Cologarithms
$$\log_{1/b} x=-\log_b x=log_b x^{-1}$$
Proof
Let $a=log_{1/b} x$ so that by
logarithmic definition and applying the
negative exponent
$$x=b^{–a}$$
Take the logarithm of base $b$
$$\log_b x= \log_b b^{-a}$$
Apply the
base inverse property
$$\log_b x=-a$$
Negate
$$-\log_b x=a$$
Substitute $a$ for $log_{1/b} x$
$$-\log_b x=log_{1/b} x$$
$–log_bx=log_bx^{–1}$ can be proven with both the
power and
quotient rules
