Pure 2 - Exponential and Logarithm

What’s an exponential function?

f(x) = a^x

(where f(x) = y)

Define increasing function (condition + description) of f(x) = a^x

Condition = a > 1

Description = as x increase, f(x) increase exponentially. As x decrease, f(x) get closer to 0

Define decreasing function (condition + description) of f(x) = a^x

Condition = a < 1 AND a > 0

Description = as x increase, f(x) get closer to 0. As x decrease, f(x) increase exponentially

Define logarithm and shows the relationship between it and exponential function

Define = an inverse exponential function

[Log_a n = x ] is equivalent to [a^x = n] (where ‘a’ is not equal to 1)

What’s Log_e x

Inverse function of e^x where e is Euler number (2.7182…)

On calculator = [In] button, input [ In(x) ]

Laws of Logarithm

1. log_aX + log_aY = log_aXY (multiplication law)

2. log_aX - log_aY = log_aX/Y (Division law)

3. log_a(X^k) = k log_aX (Power law)

List all special cases of Logarithm laws

1. log_a(1/x) = log_a(X^-1) = - log_a(X) (power law when k = -1)

2. log_a a = 1 ( a > 0 , a <> 1)

3. log_a 1 = 0 ( a > 0, a <> 1)

Why can’t this work : log₂(-4)

log_aX is an inverse function of exponential function. There’re no index/power that can make the base number turns negative

What is the “change of base” formula (and its special case)

log_ab = log_xb / log_xa ( x could be any number)

Special case: log_ab = log_bb / log_ba = 1 / log_ba

Deduce the “change of base formula”