MATH: TOPIC 6. Exponential and Logarithmic Functions

Flashcards for reviewing exponential and logarithmic functions.

Exponential Function

A function where the independent variable is an exponent.

Exponent

The independent variable in an exponential function.

'a' in Exponential Function

The factor by which f(x) changes when x increases by 1 in the function f(x) = A * a^x.

Exponential Increase Rule

If a = 1 + p/100, where p > 0 and A > 0, then f(x) will increase by p% for each unit increase in x.

Exponential Decrease Rule

If a = 1 − p/100, where 0 < p < 100 and A > 0, then f(x) will decrease by p% for each unit increase in x.

b

Vertical stretch/compression factor in the function f(x) = b ⋅ a^(d(x−c)) + k

a

Base value in the function f(x) = b ⋅ a^(d(x−c)) + k

c

Horizontal translation factor in the function f(x) = b ⋅ a^(d(x−c)) + k

d

Horizontal stretch/compression factor in the function f(x) = b ⋅ a^(d(x−c)) + k

k

Vertical translation factor in the function f(x) = b ⋅ a^(d(x−c)) + k, and it is equal to the y-value of the horizontal asymptote.

Vertical Translation

The transformation that shifts a function up or down by k units.

Horizontal Translation

The transformation that shifts a function left or right by c units.

Reflection Over X-Axis

A transformation that flips a function over the x-axis.

Reflection Over Y-Axis

A transformation that flips a function over the y-axis.

Vertical Stretch

The larger the |b| is, the more vertically stretched the graph is.

Horizontal Compression

The larger the |d| is the more horizontally compressed the graph is.

b

y-intercept (only when there is no 'c' and 'k' from the formula 6.1 . If there is no 'c' but there is 'k', the y-intercept will be b+k)

Euler's number

When a=e (e=2.7…..).

Logarithmic Function

A function in the form f(x) = log_a(x) where a>0 and a ≠ 1 and x∈(0,+∞)

Natural Logarithm (ln)

A logarithm with base 'e'.

ln

A logarithmic function which has all the properties that a normal logarithm would have.

Continuous Exponential Growth/Decay Formula

Used for tasks involving continuous growth or decay.

Exponential growth/decay Formula.

Used when something increases/decreases per unit of time/length.

1+r= 1 + 0.05 = 1.05

When something increases by 5%

1 + r = 1 + (−0.05) = 0.95

When something decreases by 5%

N(t) = N0 (1 + r)^t

A discrete exponential growth.

N(t) = N0 * e^(kt)

A continuous growth.

r in N(t) = N0 (1 + r)^t

Growth rate.

Doubling Time

The time it takes for a function to double

Half-Life

The length of time it takes for its initial quantity to decrease by half.

Lambda

The number before t, it is called the decay constant.