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4.6 Exponential and logarithmic equations  (Page 5/8)

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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 .
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Solve   ln ( x 2 ) = ln 1.

x = 1   or   x = − 1

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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
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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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