For any algebraic expressions
$\text{}S\text{}$ and
$\text{}T\text{}$ and any positive real number
$\text{}b,\text{}$ where
${b}^{S}={b}^{T}\text{}$ if and only if
$\text{}S=T.$
Definition of a logarithm
For any algebraic expression
S and positive real numbers
$\text{}b\text{}$ and
$\text{}c,\text{}$ where
$\text{}b\ne 1,$ ${\mathrm{log}}_{b}(S)=c\text{}$ if and only if
$\text{}{b}^{c}=S.$
One-to-one property for logarithmic functions
For any algebraic expressions
S and
T and any positive real number
$\text{}b,\text{}$ where
$\text{}b\ne 1,$ ${\mathrm{log}}_{b}S={\mathrm{log}}_{b}T\text{}$ if and only if
$\text{}S=T.$
Key concepts
We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.
When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown. See
[link] .
When we are given an exponential equation where the bases are
not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown. See
[link] ,
[link] , and
[link] .
When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. See
[link] .
We can solve exponential equations with base
$\text{\hspace{0.17em}}e,$ by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. See
[link] and
[link] .
After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. See
[link] .
When given an equation of the form
$\text{\hspace{0.17em}}{\mathrm{log}}_{b}(S)=c,\text{}$ where
$\text{\hspace{0.17em}}S\text{\hspace{0.17em}}$ is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation
$\text{\hspace{0.17em}}{b}^{c}=S,\text{}$ and solve for the unknown. See
[link] and
[link] .
We can also use graphing to solve equations with the form
$\text{\hspace{0.17em}}{\mathrm{log}}_{b}(S)=c.\text{\hspace{0.17em}}$ We graph both equations
$\text{\hspace{0.17em}}y={\mathrm{log}}_{b}(S)\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}y=c\text{\hspace{0.17em}}$ on the same coordinate plane and identify the solution as the
x- value of the intersecting point. See
[link] .
When given an equation of the form
$\text{\hspace{0.17em}}{\mathrm{log}}_{b}S={\mathrm{log}}_{b}T,\text{}$ where
$\text{\hspace{0.17em}}S\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}T\text{\hspace{0.17em}}$ are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation
$\text{\hspace{0.17em}}S=T\text{\hspace{0.17em}}$ for the unknown. See
[link] .
Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm. See
[link] .
Section exercises
Verbal
How can an exponential equation be solved?
Determine first if the equation can be rewritten so that each side uses the same base. If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use properties of logarithms to solve.
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
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 ?
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
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 Â that should not be there. It doesn't seem to show if encloses in quotation marks.
"Â" or 'Â' ... Â
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Period =2π
if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts