Evaluate
$\text{\hspace{0.17em}}y=\mathrm{log}(1000)\text{\hspace{0.17em}}$ without using a calculator.
First we rewrite the logarithm in exponential form:
$\text{\hspace{0.17em}}{10}^{y}=1000.\text{\hspace{0.17em}}$ Next, we ask, “To what exponent must
$\text{\hspace{0.17em}}10\text{\hspace{0.17em}}$ be raised in order to get 1000?” We know
Rewriting and solving a real-world exponential model
The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation
$\text{\hspace{0.17em}}{10}^{x}=500\text{\hspace{0.17em}}$ represents this situation, where
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?
We begin by rewriting the exponential equation in logarithmic form.
The amount of energy released from one earthquake was
$\text{\hspace{0.17em}}\text{8,500}\text{\hspace{0.17em}}$ times greater than the amount of energy released from another. The equation
$\text{\hspace{0.17em}}{10}^{x}=8500\text{\hspace{0.17em}}$ represents this situation, where
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?
The difference in magnitudes was about
$\text{\hspace{0.17em}}\mathrm{3.929.}$
The most frequently used base for logarithms is
$\text{\hspace{0.17em}}e.\text{\hspace{0.17em}}$ Base
$\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ logarithms are important in calculus and some scientific applications; they are called
natural logarithms . The base
$\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ logarithm,
$\text{\hspace{0.17em}}{\mathrm{log}}_{e}\left(x\right),$ has its own notation,
$\text{\hspace{0.17em}}\mathrm{ln}(x).$
Most values of
$\text{\hspace{0.17em}}\mathrm{ln}\left(x\right)\text{\hspace{0.17em}}$ can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base,
$\text{\hspace{0.17em}}\mathrm{ln}1=0.\text{\hspace{0.17em}}$ For other natural logarithms, we can use the
$\text{\hspace{0.17em}}\mathrm{ln}\text{\hspace{0.17em}}$ key that can be found on most scientific calculators. We can also find the natural logarithm of any power of
$\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ using the inverse property of logarithms.
Definition of the natural logarithm
A
natural logarithm is a logarithm with base
$\text{\hspace{0.17em}}e.$ We write
${\mathrm{log}}_{e}\left(x\right)$ simply as
$\mathrm{ln}\left(x\right).$ The natural logarithm of a positive number
$x$ satisfies the following definition.
We read
$\text{\hspace{0.17em}}\mathrm{ln}\left(x\right)\text{\hspace{0.17em}}$ as, “the logarithm with base
$\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ of
$\text{\hspace{0.17em}}x$ ” or “the natural logarithm of
$\text{\hspace{0.17em}}x.$ ”
The logarithm
$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ is the exponent to which
$\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ must be raised to get
$\text{\hspace{0.17em}}x.$
Since the functions
$\text{\hspace{0.17em}}y=e{}^{x}\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}y=\mathrm{ln}\left(x\right)\text{\hspace{0.17em}}$ are inverse functions,
$\text{\hspace{0.17em}}\mathrm{ln}\left({e}^{x}\right)=x\text{\hspace{0.17em}}$ for all
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}e{}^{\mathrm{ln}(x)}=x\text{\hspace{0.17em}}$ for
$\text{\hspace{0.17em}}x>0.$
Given a natural logarithm with the form
$\text{\hspace{0.17em}}y=\mathrm{ln}\left(x\right),$ evaluate it using a calculator.
Press
[LN] .
Enter the value given for
$\text{\hspace{0.17em}}x,$ followed by
[ ) ] .
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
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
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
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
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.