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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

can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
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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