ELF - Exponential and Logarithms Module Overview

Exponential and Logarithms

Introduction

The applications of exponential and logarithmic functions are vast and varied. The following are just some of the applications and jobs of exponential and logarithmic functions: savings and investment accounts, loans for cars, loans for homes, nuclear power and radioactivity, forensics, sound technician, studying earthquakes, statistician working with population growth, SAT scores, etc.

Exponents (also known powers) may be written as the number of times a base is used as a factor. They may be written as a logarithm when you need to solve for the power. The graphs of exponential and logarithmic functions show the domain, the intercepts, and the end behavior of the function. Exponential and logarithmic functions are inverse functions of each other, and the x-values and the y-values may be exchanged for each other. In modeling real-world situations, we often need to solve these functions. The steps are similar to solving other algebra equations, but usually require changing from one form to the other. They are often used to calculate things like the time to double, triple (an investment or a population), or halve (radioactive half-life) a quantity.

Essential Questions

How are exponential growth functions graphed?
How are exponential growth functions applied?
How are exponential decay functions graphed?
How are exponential decay functions applied?
How is Euler's number "e" used in exponential and logarithmic functions?
How are logarithmic functions graphed and applied?
How are logarithmic properties applied?
How are exponential and logarithmic equations solved?

Exponential and Logarithm Key Terms

The following key terms will help you understand the content in this module.

Exponential Function - A function of the form equation image indicator $LaTeX: A=a\cdot b^{x-h}+k,\:where\:a,\:h,\:and\:k$ $A=a\cdot b^{x-h}+k,\:where\:a,\:h,\:and\:k$ are real numbers, b > 0, and a and b are ≠ 1.

Exponential Growth Function - A function of the form equation image indicator $LaTeX: A=a\cdot b^{x-h}+k$ $A=a\cdot b^{x-h}+k$ where b > 1.

Growth Factor - The base number "b" with a value b > 1 in a function of the form equation image indicator $LaTeX: A=a\cdot b^{x-h}+k$ $A=a\cdot b^{x-h}+k$ where b > 1.

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.

Exponential Decay Function - A function of the form equation image indicator $LaTeX: A=a\cdot b^{x-h}+k$ $A=a\cdot b^{x-h}+k$ where 0 < b < 1.

Decay Factor - The base number "b" with a value 0 < b < 1 in a function of the form equation image indicator $LaTeX: A=a\cdot b^{x-h}+k$ $A=a\cdot b^{x-h}+k$ where 0 < b < 1.

Natural Base e - Euler's number e with the approximation of 2.718…

Common Logarithm - A logarithm with a base of 10. A common logarithm is the exponent, a, such that equation image indicator LaTeX: 10^a=b $10^a=b$ . The common logarithm of x is written log x. For example, log 100 = 2 because LaTeX: 10^2=100 $10^2=100$ .

Natural Logarithm - A logarithm with a base of e. lnb is the exponent, a, such that equation image indicator LaTeX: e^a=b $e^a=b$ . The natural logarithm of x is written lnx and represents $LaTeX: \log_ex$ $\log_ex$ . For example, ln 8 = 2.0794415… because $LaTeX: e^{2.0794415}=8$ $e^{2.0794415}=8$ .

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 equation image indicator $LaTeX: A=P\left(1+\frac{r}{n}\right)^{nt}$ $A=P\left(1+\frac{r}{n}\right)^{nt}$ 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.

Continuous Compound Interest Formula - Interest that is, theoretically, computed and added to the balance of an account each instant. The formula is equation image indicator $LaTeX: A=Pe^{rt}$ $A=Pe^{rt}$ , 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.

IMAGES CREATED BY GAVS