Exponential and Logarithm Key Terms (with images)

log_{2}8 in spoken words

"log 2 of 8"

What values are sufficient for drawing a "quick graph"?

1 and 0 (along with the asymptote)

Natural Base e

Euler's number e with the approximation of 2.718...

Exponents and logarithmic operations undo each other since they are...

inverse operations.

Change Of Base Formula

log_{a}b = (log_b)/(log_a)

x + h will give you a...

negative (left) shift

Exponential Decay Model

y = a(1 - r)^t

The asymptote (or boundary line) has the equation of...

y = k

Exponent and logarithmic functions with the same base number are represented on the graph as reflected over the...

y = x axis

log_{2}8 = 2 to the what power is 8?

3

Exponential Decay Function

A function of the form A = a * b^{x - h} + k, where 0 < b < 1.

Exponential Function

A function of the form A = a * b^{x - h} + k, where a, h, and k are real numbers, b > 0, and a and b are ≠ 1.

Exponential Growth Function

A function of the form A = a * b^{x - h} + k, where b > 1.

Common Logarithm

A logarithm with a base of 10. A common logarithm is the exponent, a, such that 10^a = b. The common logarithm of x is written log x. For example, log 100 = 2 because 10^2 = 100.

Natural Logarithm

A logarithm with a base of e. lnb is the exponent, a, such that e^a = b. The natural logarithm of x is written lnx and represents log_{e}x. For example, ln 8 = 2.0794415... because e^2.01=794415... = 8

Common Logarithms

have a base of 10. When "log x" is written, the base number of the logarithm is assumed to be "10."

h is the horizontal shift. Since the formula says x - h, a number subtracted from x gives you a...

positive (right) shift

When we are graphing an exponential function "by hand", it is best to create a table of values, and the easiest numbers to use are...

(-2, -1, 0, 1, 2)

Compound Interest Formula

A method of computing the interest, after a specified time, and adding the interest to the balance of the account. Interest can be computed as little as once a year to as many times as one would like. The formula is A = P(1 + r/n)^{nt} where A is the ending amount, P is the principal or initial amount, r is the annual interest rate, n is the number of times compounded per year, and t is the number of years.

Asymptote

An asymptote is a line or curve that approaches a given curve arbitrarily closely. A graph never crosses a vertical asymptote, but it may cross a horizontal or oblique asymptote.

Index

Another word for exponent

Graphing Exponential vs. Logarithmic example

In simple terms, an exponential function with a base number of 2 has a domain of all real numbers and a range that starts close to zero (but never touches 0) and goes "up" to positive infinity. This function will have a horizontal asymptote at y = 0. Therefore, the graph of the logarithmic functions with a base of 2 will have a domain that starts close to 0 (but never touches it) and goes "right" to infinity. Its range will be all real numbers. This function will have a vertical asymptote at x = 0.

Continuous Compound Interest Formula

Interest that is, theoretically, computed and added to the balance of an account each instant. The formula is A = Pe^{rt} where A is the ending amount, P is the principal or initial amount, r is the annual interest rate, and t is the time in years.

Decay Factor

The base number "b" with a value 0 < b < 1 in a function of the form A = a * b^{x - h} + k, where 0 < b < 1.

Growth Factor

The base number "b" with a value b > 1 in a function of the form A = a * b^{x - h} + k,where b > 1.

In exponential growth functions when a is greater than 0, the ends of the graph behave like the following:

as x goes to the left forever (negative infinity), the y (or f(x)) approaches the asymptote (y = k) from above and as x goes to the right forever (positive infinity), the y (or f(x)) goes up forever (to positive infinity).

In exponential growth functions, when a is less than 0, the ends of the graph behave like the following:

as x goes to the left forever (negative infinity), the y (or f(x)) goes down forever (to negative infinity) and as x goes to the right forever (positive infinity), the y (or f(x)) approaches the asymptote (y = k) from below.

Why do we need the Change of Base formula?

because calculators only have the common log (base 10 denoted by "log") and the natural log (base e denoted by "ln").

"b" is the base of the exponent, and if its value is between 0 and 1...

exponential decay is present.

"b" is the base of the exponent, and if its value is greater than 1...

exponential growth is present.