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 $b \neq 1$).

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 $(0, \infty)$.

Horizontal asymptote (exponential parent)

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

End behavior (growth base)

If $b > 1$, then as $x \rightarrow \infty$, $b^x \rightarrow \infty$ and as $x \rightarrow -\infty$, $b^x \rightarrow 0$.

End behavior (decay base)

If $0 < b < 1$, then as $x \rightarrow \infty$, $b^x \rightarrow 0$ and as $x \rightarrow -\infty$, $b^x \rightarrow \infty$.

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 $f(x) = a \times b^{(x-h)} + k$, if $a < 0$ the graph is reflected over the horizontal line $y=k$.

Parameter h (horizontal shift)

In $f(x) = a \times b^{(x-h)} + k$, $h$ shifts the graph right by $h$ (because the exponent is $x-h$).

Parameter k (vertical shift)

In $f(x) = a \times b^{(x-h)} + k$, $k$ shifts the graph up/down and moves the horizontal asymptote to $y = k$.

Horizontal asymptote (transformed exponential)

For $f(x) = a \times b^{(x-h)} + k$, the horizontal asymptote is $y = k$.

Range of a transformed exponential

For $f(x) = a \times b^{(x-h)} + k$: 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 $f(x) = a \times b^x$, the constant ratio over 1 unit is $\frac{f(x+1)}{f(x)} = b$.

Percent growth factor

If a quantity increases by $r\\%$ per period, the exponential factor is $b = 1 + \frac{r}{100}$.

Percent decay factor

If a quantity decreases by $r\%$ per period, the exponential factor is $b = 1 - \frac{r}{100}$.

Doubling time model

If a quantity doubles every $d$ time units: $A(t) = A_0 \times 2^{(t/d)}.$

Half-life model

If a quantity has half-life $h$: $A(t) = A_0 \times (1/2)^{(t/h)}$.

Discrete-time exponential model

$A(t) = A_0 \times b^t$, used when change occurs in equal steps (each period multiplies by $b$).

Continuous-time exponential model

$A(t) = A_0 \times e^{(kt)}$, used when change occurs continuously; growth/decay depends on the sign of $k$.

Compound interest (discrete compounding)

$A(t) = P(1 + \frac{r}{n})^{(nt)}$, 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

$\log_b(x) = \frac{\ln(x)}{\ln(b)}$ (or $\frac{\log(x)}{\log(b)}$), used to compute logs with unsupported bases.

Power rule of logarithms

$\log_b(M^p) = p \times \log_b(M)$, which brings down an exponent as a multiplier.