1. Home
  2. Precalculus
  3. Exponential and logarithmic
  4. Exponential and logarithmic

4.6 Exponential and logarithmic equations  (Page 5/8)

<< Chapter < Page
  Precalculus     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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir 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, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask