Step-by-step explanation:
The first technique we will introduce for solving exponential equations involves two functions with like bases. Recall that the one-to-one property of exponential functions tells us that, for any real numbers b, S, and T, where
b
>
0
,
b
≠
1
,
b
S
=
b
T
if and only if S = T.
In other words, when an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, 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.
For example, consider the equation
3
4
x
−
7
=
3
2
x
3
. To solve for x, we use the division property of exponents to rewrite the right side so that both sides have the common base 3. Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for x:
3
4
x
−
7
=
3
2
x
3
3
4
x
−
7
=
3
2
x
3
1
Rewrite 3 as 3
1
.
3
4
x
−
7
=
3
2
x
−
1
Use the division property of exponents
.
4
x
−
7
=
2
x
−
1
Apply the one-to-one property of exponents
.
2
x
=
6
Subtract 2
x
and add 7 to both sides
.
x
=
3
Divide by 3
.
A GENERAL NOTE: USING THE ONE-TO-ONE PROPERTY OF EXPONENTIAL FUNCTIONS TO SOLVE EXPONENTIAL EQUATIONS
For any algebraic expressions S and T, and any positive real number
b
≠
1
,
b
S
=
b
T
if and only if
S
=
T
HOW TO: GIVEN AN EXPONENTIAL EQUATION OF THE FORM
b
S
=
b
T
, WHERE S AND T ARE ALGEBRAIC EXPRESSIONS WITH AN UNKNOWN, SOLVE FOR THE UNKNOWN
Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form
b
S
=
b
T
.
Use the one-to-one property to set the exponents equal to each other.
Solve the resulting equation, S = T, for the unknown.
The record low temperature in Houston for November 12th is -3 degrees. The record high temperature is 83 degrees. What is the difference between the record high temperature and the record low temperature?
Answer:
80 degrees
Step-by-step explanation:
-3 + 83 is the same as 83 -3.
Please help me with this question
Answer:
64%
Step-by-step explanation:
1. First, we must know that the fraction 48/75 represents that out of 75 total fish, 48 of them were trout.
2. Now, let's divide 48 by 75:
48 ÷ 750.643. Lastly, to convert it into a percentage, we must multiply 0.64 by 100:
0.64 × 1006464%Therefore, the answer is 64%.
what is c in c/4 = - 9/8 ?
Answer:
c = -9/2
Step-by-step explanation:
4. A colony of bacteria is growing in such a way that each generation is 2.5 times as large as the previous generation. Suppose there are 450 bacteria in the first generation.
a) Show that nº generation is 180(2.5)^n
b) How many will there be in the fifth generation?
c) What will be the total number of bacteria in the first tenth generations to the nearest integer? [10 marks]
Step-by-step explanation:
I think there are some typos in a)
anyway, this is a geometric sequence.
that means that after a starting value each new term is created by multiplying the previous term by a certain factor.
a)
s1 = 450
factor f = 2.5
so,
s2 = 450×2.5 = 1125
and so on.
from that we see that
sn = 450×(2.5)^(n-1)
and as
450/2.5 = 180
or
450 = 180×2.5
we can "extract" another 2.5 factor from the basic 450.
so,
sn = 180×2.5×(2.5)^(n-1) = 180×(2.5)^n
b)
s5 = 180×(2.5)⁵ = 17,578.125 or rounded 17578
c)
the question is unclear. are the bacteria of previous generations still there, when a new one is generated ?
then the 2.5 factor creates the sum implicitly, and the total number of bacteria in the first 10 generations is simply the 10th term s10.
s10 = 180×(2.5)¹⁰ = 1716613.76953125 = 1,716,614
or is it meant to sum the numbers of each generation (the old bacteria "go away", when the next generation is generated) ?
the sum of the first n terms of such a geometric sequence is
sum = a1×(1 - f^n)/(1 - f)
so, for our first ten terms
sum = 450×(1 - 2.5^10)/(1 - 2.5) =
= 450×-9535.7431640625/-1.5 =
2860722.94921875 = rounded 2,860,723
0.57356 multiplied by 10 equals______ (with solution)
Answer:
5.7356
Step-by-step explanation:
please mark me as brainlest and
Answer:
0.57356 × 10 = 5.7356
Step-by-step explanation:
in multiplying any decimal to number 10.. the easiest trick to do here is to move the decimal point to the right. since we have number 10 and it has one 0's in it ...
...we only move the decimal point to the right one time!so your answer to your question is 5.7356Answer the question in the photo please.
Answer:
The answer is -2
Step-by-step explanation:
2 to the 3rd power would be the same as saying 2x2x2 which is 8. Now with that, you know that y is 6 and o is 4, so you would do 8-6-4. I hope this helps!
One taxi company
charges $5 just to
stop, and $0.8 per
mile after that.
Another taxi company
charges $11 just to
stop, and $0.6 per
mile after that.
For _____miles the
two companies will
cost the same.
I cant solve this pls help
Answer:
x is element of -1/2 and 2
Step-by-step explanation:
this is the answer
brainlist please
Write in words 572, 198.
Answer:
five hundred and seventy two, one hundred and ninety eight
Step-by-step explanation:
What is one sixth of 192?
. Write the equation of a line with a y-intercept of 2 and a slope of - Graph the equation 3
Answer:
y=3/4x+2
Step-by-step explanation:
Idont know how to graph on brainly
What is the recursive formula for this sequence?
10, 14, 18, 22, 26,...
O A. (
= 4
len = 2n-1 +10
=
O B.
ay = 30
an = an-1 + 4
=
C.
(ay = 10
lan= any +4
O D.
ſay = 10
Lan = an-1-4
lan
Answer:
[tex]\begin{cases}a_1 = 10\\a_{n+1} = a_n + 4\end{cases}[/tex]
==================================================
Explanation:
The first row [tex]a_1 = 10[/tex] indicates the first term is 10. The small "1" is the index number. That means [tex]a_2[/tex] is next followed by [tex]a_3[/tex] and so on.
To generate the next term, we follow this recursive step: [tex]a_{n+1} = a_n + 4[/tex]
It means "whatever the nth term [tex]a_n[/tex] is, we add 4 to it to get the next term [tex]a_{n+1}[/tex]"
In other words, we add 4 to each term to get the next term.
Eg: 10+4 = 14 and 14+4 = 18
The recursive step could be rewritten as [tex]a_n = a_{n-1}+4[/tex] based on how you frame things.
A student measured the length of a certain bar as 20 cm, which exceeded the actual length by 10% the actual length of the bar was
Answer:
18cm
Step-by-step explanation:
20/10 = 2cm
So since 2x10 is 20 and it exceeded only 10%, we only subtract 2 from 20 which you leave you with 18cm.
Answer the question in the photo please
Answer:
Brian because if you do the math correctly it should be Brian
Answer:
Laurie
Step-by-step explanation:
Laurie
12 +3 (3² + (-9)) = 12 +3 (9 +(-9)) {Here, 3² is calculated}
= 12 + 3 *0 {9 - 9 = 0}
= 12 + 0 {Anything multiplied by 0 = 0}
= 12
Simplify the following polynomial expression.
(3x^2 – 1x– 7) - (532 – 4x - 2) + (c+ 3)(x+ 2)
Answer:
cx+3x^2+2c+6x-531
Step-by-step explanation:
What is the measure of the missing exterior angle?
Answer:
54 degrees
Step-by-step explanation:
The sum of all the exterior angles on ANY convex polygon = 360 degrees
X = 360-47-97-36-88-38
= 54 degrees
Which of the following place values would be found to the left of the 1 in 0.01?
1 ones
2 tenths
3 hundredths
4 thousandths
Answer:
tenths
Step-by-step explanation:
Tenths would be the place value found to the left of the 1 in 0.01 because the one is in the hundredths place value. If you move over to the left, it would be the tenths. Hope this helps!
CAN SOMEONE HELP ME EVALUATE THE EXPRESSION (4^2-4)+6/2
Answer:
2 × [tex]4x^{n}[/tex]- 1
Step-by-step explanation:
please help I'm gonna fail
Answer:
Ask me things ill try my best!
5−14x=15−14x. help me please .........................................................................
x : (x + 3) = 2 : 3
Work out the value of x
Step-by-step explanation:
x : (x + 3) = 2 : 3
[tex] \frac{x}{x + 3} = \frac{2}{3} [/tex]
[tex]3x = 2x + 6[/tex]
[tex]3x - 2x = 6[/tex]
[tex]x = 6[/tex]
The value of x on simplifying the expression x : (x + 3) = 2 : 3 is 6.
What is the ratio?If b is not equal to 0, an ordered pair of numbers a and b, denoted as a / b, is a ratio. A proportion is an equation that equalizes two ratios. It is the contrast of two different quantities. For instance, if your father weighs 90 kg, and you weigh 30 kg, the weight ratio is 1:3.
Given:
x : (x + 3) = 2 : 3,
Calculate the value by simplifying the expression,
3x = 2 (x + 3)
3x = 2x + 6
x = 6
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2014, a toy store sold 12,824 stuffed animals. In 2015, the toy store sold 24,490 more stuffed animals than it did in 2014. The toy store sold the same number of stuffed animals in 2016 as it did in 2015. How many stuffed animals did the toy store sell in 2014, 2015, and 2016 combined?
Answer:
87452
Step-by-step explanation:
not needed
Answer: in 2015 they sold 37,314. In total they sold 87,452 in all three years combined.
Step-by-step explanation:
Baby Unicorns have a weekly training schedule, flying for 4 hours per day on some days and 2 hours per day on the other days. They fly a total of 24 hours in a seven-day week. On how many days do they fly for 4 hours?
The Baby unicorns fly 4 hours for 5 days and 2 hours for 2 days.
Let x represent the number of days flown 4 hours and y represent the number of days flown 2 hours
Since they worked a total of 7 days, hence:
x + y = 7 (1)
Also, they flew for 24 hours hence:
4x + 2y = 24 (2)
Solving equations 1 and 2 simultaneously gives:
x = 5, y = 2
Therefore they fly 4 hours for 5 days and 2 hours for 2 days.
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Please help me out please
9514 1404 393
Answer:
(a) m = 0.47, b = 154.30
(b) b = fixed costs, listed as insurance and car payments.
Step-by-step explanation:
(a) All you're required to do here is recognize that the fixed costs of $154.30 for insurance and car payments represent the constant part of the monthly expense. And, the $0.47 per mile represents the mileage-related part of the monthly expense. Of course the mileage-related charge is the product of the per-mile charge ($0.47) and the number of miles (x).
Then the monthly charge is ...
monthly cost = mileage-related cost + constant (fixed) cost
y = mx + b
y = 0.47x + 154.30 ⇔ m = 0.47, b = 154.30
__
(b) The 'b' in the equation is the fixed monthly cost: $154.30 for insurance and car payment.
Suppose that g(x) = f(x) + 6. Which statement best compares the graph of
g(x) with the graph of f(x)?
Answer:
The graph of the function is shifted 6 units above that of the graph of f.
Step-by-step explanation:
Assuming that f and g are functions that map real numbers to real numbers, the graph of f is the set
{f(x) : x in R }
and the graph of g is the set
{g(x) : x in R}
where R is the set of real numbers. Since g(x) = f(x), the graph of g is then
{f(x) + 6 : x in R}.
3.
If the second and the seventh terms of a geometric progression are 6 and 1458
respectively, find the first term.
Answer:
steps below
Step-by-step explanation:
aₙ = a₁ * rⁿ⁻¹
a₁ * r²⁻¹ = 6
a₁ * r = 6 ... (1)
a₁ * r⁵ = 1458 ... (2)
(2)/(1) : r⁴ = 243 r = ±3 [tex]\sqrt[4]{3\\}[/tex]
if r = 3 [tex]\sqrt[4]{3\\}[/tex]
[tex]a_{1} = 6 / 3\sqrt[4]{3} \\a_{1} = \frac{2}{3} \sqrt[4]{27}[/tex]
if r = -3 [tex]\sqrt[4]{3\\}[/tex]
[tex]a_{1} = -\frac{2}{3} \sqrt[4]{27}[/tex]
The first term of the geometric progression is 256.
What is a geometric sequence?A geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant.
We have,
Let the first term of the geometric progression be denoted by 'a', and let the common ratio be denoted by 'r'.
According to the problem, the second term is 6, so we have:
ar = 6 ...(1)
Similarly, the seventh term is 1458, so we have:
[tex]ar^6[/tex] = 1458 ...(2)
We can use equation (1) to solve for 'r', and then substitute the value of 'r' into equation (2) to solve for 'a'.
From equation (1), we have:
r = 6/a
Substituting this into equation (2), we get:
[tex]a(6/a)^6[/tex] = 1458
Simplifying the right side of the equation, we have:
[tex]6^6[/tex] = 1458a
Solving for 'a', we get:
a = [tex]6^6[/tex]/1458 = 256
Therefore,
The first term of the geometric progression is 256.
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WHERE DO THEY GO??? PLEASE HELP MEEE
Answer:
Step 1: 2x - 6 = 4x
Step 2: -6 = 2x
Step 3: -3 = x
Step-by-step explanation:
Step 1: Start off with the distributive property, multiplying 2 by x and -3.
2(x - 3) = 4x ------> 2x - 6 = 4x
Step 2: Subtract 2x from both sides.
2x - 6 = 4x ------> -6 = 2x
Step 3: Divide by 2 to solve for x.
-6 = 2x ------> x = -3
ngl you should probably know this...
Answer:
4, 3, then 5
Step-by-step explanation:
1. 2x-6=4x
2. -6=2x
3. -3=x
20mnx^{2} \frac{x}{y} 50mx^{2} 4nx^{2} 9
Answer: 20m^(2)-9m-20
Step-by-step explanation
Harry cycles 8 kilometres in 30 minutes.
Calculate his average speed, in km/h
Answer:
16km/h
Step-by-step explanation:
average speed = total distance traveled / total time taken
30 min = 0.5 h
8/0.5 = 16km/h
Cassandra Lu earns $433.21 a week. The Social Security tax rate is 6.2% of
gross pay. The Medicare tax is 1.45% of gross pay. How much are her paycheck
deductions this week for Social Security and Medicare tax?
A.$33.99
B.$33.14
C.$34.43
D.$35.25
Work Shown:
Social Security = 6.2% of 433.21 = 0.062*433.21 = 26.86
Medicare = 1.45% of 433.21 = 0.0145*433.21 = 6.28
Add up those results to get 26.86+6.28 = 33.14