1. Home
  2. College algebra
  3. Exponential and logarithmic
  4. Exponential and logarithmic

6.6 Exponential and logarithmic equations  (Page 5/8)

<< Chapter < Page
  College algebra     Page 5 / 8
Chapter >> Page >

To check the result, substitute   x = 10   into   log ( 3 x − 2 ) − log ( 2 ) = log ( x + 4 ) .

log ( 3 ( 10 ) − 2 ) − log ( 2 ) = log ( ( 10 ) + 4 )              log ( 28 ) − log ( 2 ) = log ( 14 )                             log ( 28 2 ) = log ( 14 ) The solution checks .

Using the one-to-one property of logarithms to solve logarithmic equations

For any algebraic expressions   S   and   T   and any positive real number   b , where   b ≠ 1 ,

log b S = log b T     if and only if     S = T

Note, when solving an equation involving logarithms, always check to see if the answer is correct or if it is an extraneous solution.

Given an equation containing logarithms, solve it using the one-to-one property.

  1. Use the rules of logarithms to combine like terms, if necessary, so that the resulting equation has the form   log b S = log b T .
  2. Use the one-to-one property to set the arguments equal.
  3. Solve the resulting equation,   S = T , for the unknown.

Solving an equation using the one-to-one property of logarithms

Solve   ln ( x 2 ) = ln ( 2 x + 3 ) .

                           ln ( x 2 ) = ln ( 2 x + 3 )                               x 2 = 2 x + 3 Use the one-to-one property of the logarithm .            x 2 − 2 x − 3 = 0 Get zero on one side before factoring .       ( x − 3 ) ( x + 1 ) = 0 Factor using FOIL .                      x − 3 = 0  or  x + 1 = 0 If a product is zero, one of the factors must be zero .                               x = 3  or  x = − 1 Solve for  x .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve   ln ( x 2 ) = ln 1.

x = 1   or   x = − 1

Got questions? Get instant answers now!

Solving applied problems using exponential and logarithmic equations

In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve logarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm.

One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life . [link] lists the half-life for several of the more common radioactive substances.

Substance Use Half-life
gallium-67 nuclear medicine 80 hours
cobalt-60 manufacturing 5.3 years
technetium-99m nuclear medicine 6 hours
americium-241 construction 432 years
carbon-14 archeological dating 5,715 years
uranium-235 atomic power 703,800,000 years

We can see how widely the half-lives for these substances vary. Knowing the half-life of a substance allows us to calculate the amount remaining after a specified time. We can use the formula for radioactive decay:

A ( t ) = A 0 e ln ( 0.5 ) T t A ( t ) = A 0 e ln ( 0.5 ) t T A ( t ) = A 0 ( e ln ( 0.5 ) ) t T A ( t ) = A 0 ( 1 2 ) t T

where

  • A 0   is the amount initially present
  • T   is the half-life of the substance
  • t   is the time period over which the substance is studied
  • y   is the amount of the substance present after time   t

Using the formula for radioactive decay to find the quantity of a substance

How long will it take for ten percent of a 1000-gram sample of uranium-235 to decay?

          y = 1000 e ln ( 0.5 ) 703,800,000 t       900 = 1000 e ln ( 0.5 ) 703,800,000 t After 10% decays, 900 grams are left .        0.9 = e ln ( 0.5 ) 703,800,000 t Divide by 1000 . ln ( 0.9 ) = ln ( e ln ( 0.5 ) 703,800,000 t ) Take ln of both sides . ln ( 0.9 ) = ln ( 0.5 ) 703,800,000 t ln ( e M ) = M              t = 703,800,000 × ln ( 0.9 ) ln ( 0.5 ) years Solve for  t .              t ≈ 106,979,777 years
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

what is biology
Hajah Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
what is biology
Victoria Reply
HOW CAN MAN ORGAN FUNCTION
Alfred Reply
the diagram of the digestive system
Assiatu Reply
allimentary cannel
Ogenrwot
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 1

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask