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log 2 ( 2 ) + log 2 ( 3 x 5 ) = 3             log 2 ( 2 ( 3 x 5 ) ) = 3 Apply the product rule of logarithms .                log 2 ( 6 x 10 ) = 3 Distribute .                                 2 3 = 6 x 10 Apply the definition of a logarithm .                                    8 = 6 x 10 Calculate  2 3 .                                  18 = 6 x Add 10 to both sides .                                   x = 3 Divide by 6 .

Using the definition of a logarithm to solve logarithmic equations

For any algebraic expression S and real numbers b and c , where b > 0 ,   b 1 ,

log b ( S ) = c if and only if b c = S

Using algebra to solve a logarithmic equation

Solve 2 ln x + 3 = 7.

2 ln x + 3 = 7       2 ln x = 4 Subtract 3 .         ln x = 2 Divide by 2 .             x = e 2 Rewrite in exponential form .
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Solve 6 + ln x = 10.

x = e 4

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Using algebra before and after using the definition of the natural logarithm

Solve 2 ln ( 6 x ) = 7.

2 ln ( 6 x ) = 7    ln ( 6 x ) = 7 2 Divide by 2 .          6 x = e ( 7 2 ) Use the definition of  ln .            x = 1 6 e ( 7 2 ) Divide by 6 .
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Solve 2 ln ( x + 1 ) = 10.

x = e 5 1

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Using a graph to understand the solution to a logarithmic equation

Solve ln x = 3.

ln x = 3 x = e 3 Use the definition of the natural logarithm .

[link] represents the graph of the equation. On the graph, the x -coordinate of the point at which the two graphs intersect is close to 20. In other words e 3 20. A calculator gives a better approximation: e 3 20.0855.

Graph of two questions, y=3 and y=ln(x), which intersect at the point (e^3, 3) which is approximately (20.0855, 3).
The graphs of y = ln x and y = 3 cross at the point (e 3 , 3 ) , which is approximately (20.0855, 3).
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Use a graphing calculator to estimate the approximate solution to the logarithmic equation 2 x = 1000 to 2 decimal places.

x 9.97

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

As with exponential equations, we can use the one-to-one property to solve logarithmic equations. The one-to-one property of logarithmic functions tells us that, for any real numbers x > 0 , S > 0 , T > 0 and any positive real number b , where b 1 ,

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

For example,

If   log 2 ( x 1 ) = log 2 ( 8 ) , then  x 1 = 8.

So, if x 1 = 8 , then we can solve for x , and we get x = 9. To check, we can substitute x = 9 into the original equation: log 2 ( 9 1 ) = log 2 ( 8 ) = 3. In other words, when a logarithmic equation has the same base on each side, the arguments must be equal. This also applies when the arguments are algebraic expressions. Therefore, when given an equation with logs of the same base on each side, we can use rules of logarithms to rewrite each side as a single logarithm. Then we use the fact that logarithmic functions are one-to-one to set the arguments equal to one another and solve for the unknown.

For example, consider the equation log ( 3 x 2 ) log ( 2 ) = log ( x + 4 ) . To solve this equation, we can use the rules of logarithms to rewrite the left side as a single logarithm, and then apply the one-to-one property to solve for x :

log ( 3 x 2 ) log ( 2 ) = log ( x + 4 )              log ( 3 x 2 2 ) = log ( x + 4 ) Apply the quotient rule of logarithms .                      3 x 2 2 = x + 4 Apply the one to one property of a logarithm .                      3 x 2 = 2 x + 8 Multiply both sides of the equation by  2.                               x = 10 Subtract 2 x  and add 2 .

Questions & Answers

what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
Practice Key Terms 1

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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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