Algebra II Final Exam - Exponential Functions and Logarithms

Basic Formula for Exponential Functions

An exponential function is defined by the
formula f(x) = a^x, where the input variable
x occurs as an exponent. The exponential
curve depends on the exponential function
and it depends on the value of the x.

Transformations

Transformations include vertical shifts,
horizontal shifts, and graph reversals.
Changing the sign of the exponent will
result in a graph reversal or flip. A positive
exponent has the graph heading to infinity
as x gets bigger. A negative exponent has
the graph heading to infinity as x gets
smaller.

Domain and Range

The domain of a function is the set of input
or x -values for which the function is
defined, while the range is the set of all the
output or y -values that the function takes.

Determining Equations Given Two Points

Step 1: Identify two points on the graph,

Step 2: Find the slope between the two points found
in step 1 using the formula. Simplify completely.

Step 3: Set up a function using the slope from step 2.
Solve for the -intercept by substituting one of the
points from step 1 into the function.

Step 4: Write the function using the slope and
-intercept found in steps 2 and 3.

Graphs

An exponential graph is a curve that has a
horizontal asymptote and it either has an
increasing slope or a decreasing slope. It
starts as a horizontal line and then it first
increases/decreases slowly and then the
growth/decay becomes rapid.

Growth and Decay

It's exponential growth when the base of
our exponential is bigger than 1, which
means those numbers get bigger. It's
exponential decay when the base of our
exponential is in between 1 and 0 and
those numbers get smaller. An asymptote
is a value that a function will get infinitely
close to, but never quite reach.

Solving Exponentials

To solve exponential equations without
logarithms, you need to have equations with
comparable exponential expressions on either
side of the "equals" sign, so you can compare the
powers and solve. In other words, you have to
have "(some base) to (some power) equals (the
same base) to (some other power)", where you
set the two powers equal to each other, and
solve the resulting equation.

Graphs of Logarithms

To graph a logarithmic function 𝑦=𝑙𝑜𝑔𝑏(𝑥) ,
it is easiest to convert the equation to its
exponential form, 𝑥=𝑏𝑦 . Generally, when
graphing a function, various 𝑥 -values are
chosen and each is used to calculate the
corresponding 𝑦 -value. In contrast, for this
method, it is the 𝑦 -values that are chosen
and the corresponding 𝑥 -values that are
then calculated.

Properties of Logarithms

The properties of log are used to expand a
single logarithm into multiple logarithms
(or) compress multiple logarithms into a
single logarithm. A logarithm is just
another way of writing exponents. Thus,
the properties of logarithms are derived
from the properties of exponents.

Change of Base

The change of base formula, as its name
suggests, is used to change the base of a
logarithm. We might have noticed that a
scientific calculator has only "log" and "ln"
buttons. Also, we know that "log" stands for a
logarithm of base 10 and "ln" stands for a
logarithm of base e. But there is no option to
calculate the logarithm of a number with any
other bases other than 10 and e.

Expanding and Condensing Logarithms

To expand logarithms, write them as a sum
or difference of logarithms where the
power rule is applied if necessary. Often,
using the rules in the order quotient rule,
product rule, and then power rule will be
helpful. The reverse process of expanding
logarithms is called combining or
condensing logarithmic expressions into a
single quantity