$log_b(a) = c$

• Logarithm is a function that represents the exponent $(c)$ or power to which a specific base number $(b)$ must be raised to obtain another term $(a)$.
• The question it seeks to answer: “To what power must I raise a given base to get a particular result?”

$log_b(a) = c <=> b^c = a$

Where

• $b$ = base
• $c$ = exponent
• $a$ = argument

$2^x = 16, x = 4$

• The base must be positive because it is an exponential function. Having a negative number for the base will result in complex functions. $(-3)^\frac{1}{2}$ = $\sqrt{-3} = 3i$.
• Also, 0 raised to any power will always result in the value of 0, making a constant function. Therefore, the base cannot be 0 because it violates properties of exponential functions.

If the base is 1, the value will always be 1 because it is just a repeated multiplication of the base it self.

For example, $log_1(3) = x$ , which is the same as $1^x = 3$, which is not true because anything to the power of 1 will always result in 1.

If $b > 0$, then $a$ must be greater than 0 because raising a positive base $b$ to any exponent will always yield a positive result.

Product Rule: $\log_b(xy) = log_b(x) +log_b(y)$

Example: $\log_2(2\cdot4) = \log_2(2) + \log_2(4) =3$

Quotient Rule: $\log_b(\frac{x}{y}) = log_b(x) - log_b(y)$

Example: $\log_3(\frac{9}{3}) = \log_3(9) - \log_3(3) =1$

Power Rule: $\log_b(x^n) = n\cdot \log_b(x)$

Example:

$\log_2(2^2) = \log_2(4) = 2$

$2\cdot\log_2(2)= 2\cdot1=2$

Equality Rule: $\log_a(x) = log_a(y), \text{ then } x=y$

$\log_a(x)=\frac{log_b(x)}{log_b(a)}$

Example:

• $\log_3(27)=3$
• $\frac{log_3(27)}{log_3(3)} = 3$

• Regular notation $\log_{10}(x)$
• Special notation $\log(x)$

• Regular notation $\log_{e}(x)$
• Special notation $\ln(x)$

$$ e = \lim_{n\rightarrow\infty}(1+\frac{1}{n})^n $$

As $n$ approaches to infinity, the value will approach to 2.718…, which is $e$, the maximum amount we get after 100% continuous compounding growth for one time period.

Similar to $\pi$, its digit go on forever without repeating.

$e$ represents continuous growth.

• Any function that has a constant base $(b)$ with an independent variable $(x)$ that is an exponent.
• $y = ab^x$, where $b > 0$

1. $b>1$, exponential increase at a growth rate of b (exponential growth).
2. $b<1$, exponential decrease at a growth rate of b (exponential decay).
3. $b =1$, $y = a$ , which is a constant value.

• Exponent with function is when the constant is an exponent. $y = x^2$
• Exponential function is when the independent variable is the exponent. $y = ab^x$

• The expression represents exponential growth or decay of quantity over time. Where $r$ is the growth rate and $t$ is the number of time periods.
• Use-Case:
  • It lets you predict the impact of any growth rate and time period, given maximum compounding growth
  • In other words, it answers how much maximum growth do I get after $t$ units of time, given the growth rate $r$.

1. $f(x)$ is the number of people on the network, a dependent variable
2. $x$ is the number of time periods (months), an independent variable
3. $a$ is a constant that represents the initial number of people using the network.
4. $b$ is the constant that is being repeatedly multiplied each step. A larger $b$ will increase the exponential effect.

$$ \ln(x) <=> log_e(x) $$

$\ln(x)$ seeks to determine the time needed to grow or shrink to $x.$ In other words, it tells you the time it takes for something to grow /shrink continuously.

• $e^x$ calculates the maximum growth over a time period.
  • Input: Time Required → Output: Required Growth
• $\ln(x)$ calculates the time required to reach maximum amount of expected growth.
  • Input: Expected Growth → Output: Required Time.

$$ \ln(e) = \log_e(e) $$ $$ \log_e(e) = a $$ $$ e^a = e $$ $$ a = 1 $$ $$ ln(e) = 1 $$

$$ e^{\ln(x)}= x $$ $$ \ln({e^{\ln(x)}}) = \ln(x) $$ $$ \ln(x)\cdot {\ln(e)} = \ln(x) $$ $$ \ln(x)\cdot 1 = \ln(x) $$ $$ \ln(x) = \ln(x) $$ $$ x=x $$

$$ y=e^x $$ $$ \ln(y) = \ln(e^x) $$ $$ \ln(y) = x $$

Where, $\ln(e^x) = x$

• $e^3=20.208$: It takes 3 units of time to get to 20x times what we started with.
• $\ln(20.08) =3$: To get to 20x times what we started with, we need 3 units of time.

$10e^{(-1)*(3)} = 0.498$

$120e^{(0.05)*(10)} = 197.84$

Not possible because growth cannot be negative.