Unit 2: Exponential and Logarithmic Functions

Sequence

An ordered list of numbers in which each number is called a term; can be finite or infinite.

Term (of a sequence)

An individual number in a sequence, identified by its position.

Finite sequence

A sequence with a last term (a limited number of terms).

Infinite sequence

A sequence that continues without end (no last term).

Arithmetic sequence

A sequence in which each successive term changes by a constant additive amount.

Common difference (d)

The constant amount added (or subtracted) to get from one term to the next in an arithmetic sequence.

Geometric sequence

A sequence in which each successive term changes by a constant multiplicative factor.

Common ratio (r)

The constant factor multiplied each step to get from one term to the next in a geometric sequence.

Constant difference test

A way to recognize linear/arithmetic behavior: over equal input steps, outputs have (approximately) constant differences.

Constant ratio test

A way to recognize exponential/geometric behavior: over equal input steps, outputs have (approximately) constant ratios.

Linear relationship (from patterns)

A model suggested when outputs change at a constant additive rate over equal input intervals.

Exponential relationship (from patterns)

A model suggested when outputs change at a constant proportional (multiplicative) rate over equal input intervals.

Exponential function

A function in which the variable appears in the exponent, often written in parent form f(x)=b^x.

Base (b)

The positive constant in b^x that determines the growth/decay factor per 1-unit increase in x (with b>0 and b≠1).

Parent exponential function

The basic exponential form f(x)=b^x (before transformations).

Power function

A function where the variable is in the base, e.g., g(x)=x^2, not in the exponent.

Exponential growth

Behavior of b^x when b>1; outputs increase as x increases.

Exponential decay

Behavior of b^x when 0<b<1; outputs decrease as x increases.

Exponential anchor point (0,1)

For any valid base b, b^0=1, so the graph of y=b^x passes through (0,1).

Exponential anchor point (1,b)

Since b^1=b, the graph of y=b^x passes through (1,b).

Domain of an exponential function

For f(x)=b^x, the domain is all real numbers.

Range of an exponential function

For f(x)=b^x, outputs are always positive, so the range is (0,∞).

Horizontal asymptote (exponential parent)

For f(x)=b^x, the horizontal asymptote is y=0.

End behavior (growth base)

If b>1, then as x→∞, b^x→∞ and as x→−∞, b^x→0.

End behavior (decay base)

If 0<b<1, then as x→∞, b^x→0 and as x→−∞, b^x→∞.

Monotonic (exponential parent)

For f(x)=b^x, the function is always increasing (b>1) or always decreasing (0<b<1), so it has no local extrema.

Concavity of the parent exponential

The parent exponential y=b^x is always concave up.

Transformed exponential model

A common form f(x)=a·b^(x−h)+k used to model real situations with shifts and scaling.

Parameter a (in a·b^(x−h)+k)

Controls vertical stretch/compression and reflection; negative a reflects the graph across y=k.

Vertical stretch/compression

If |a|>1, the graph stretches vertically; if 0<|a|<1, it compresses vertically.

Vertical reflection across y=k

In f(x)=a·b^(x−h)+k, if a<0 the graph is reflected over the horizontal line y=k.

Parameter h (horizontal shift)

In f(x)=a·b^(x−h)+k, h shifts the graph right by h (because the exponent is x−h).

Parameter k (vertical shift)

In f(x)=a·b^(x−h)+k, k shifts the graph up/down and moves the horizontal asymptote to y=k.

Horizontal asymptote (transformed exponential)

For f(x)=a·b^(x−h)+k, the horizontal asymptote is y=k.

Range of a transformed exponential

For f(x)=a·b^(x−h)+k: if a>0 then f(x)>k; if a<0 then f(x)<k.

Exponent shift identity

b^(x+k)=b^x·b^k, so adding inside the exponent can be rewritten as multiplying by a constant.

Exponent scaling identity

b^(cx)=(b^c)^x, so scaling x in the exponent changes the effective base.

Multiplicative rate of change (factor per step)

For f(x)=a·b^x, the constant ratio over 1 unit is f(x+1)/f(x)=b.

Percent growth factor

If a quantity increases by r% per period, the exponential factor is b=1+r/100.

Percent decay factor

If a quantity decreases by r% per period, the exponential factor is b=1−r/100.

Doubling time model

If a quantity doubles every d time units: A(t)=A0·2^(t/d).

Half-life model

If a quantity has half-life h: A(t)=A0·(1/2)^(t/h).

Discrete-time exponential model

A(t)=A0·b^t, used when change occurs in equal steps (each period multiplies by b).

Continuous-time exponential model

A(t)=A0·e^(kt), used when change occurs continuously; growth/decay depends on the sign of k.

Compound interest (discrete compounding)

A(t)=P(1+r/n)^(nt), where n is the number of compounding periods per year and nt is total periods.

Continuous compounding

Often written A(t)=Pe^(rt), modeling interest compounded continuously.

Discrete–continuous rate link

The discrete factor and continuous rate satisfy b=e^k and k=ln(b).

Logarithm (definition)

log_b(x)=y means b^y=x; the log gives the exponent needed on base b to produce x.

Domain of a logarithm

For f(x)=log_b(x), inputs must satisfy x>0.

Change of base formula

log_b(x)=ln(x)/ln(b) (or log(x)/log(b)), used to compute logs with unsupported bases.

Power rule of logarithms

logb(M^p)=p·logb(M), which brings down an exponent as a multiplier.