A group of related facts that use the same numbers is an

Please help! Its for my big test tomorrow!

Travka [436]

Answer:

# The solution x = -5

# The solution is x = 1

# The solution is x = 6.4

# The solution is x = 4

# The solution is 1.7427

# The solution is 0.190757

Step-by-step explanation:

* Lets revise some rules of the exponents and the logarithmic equation

# Exponent rules:

1- b^m × b^n = b^(m + n) ⇒ in multiplication if they have same base

we add the power

2- b^m ÷ b^n = b^(m – n) ⇒ in division if they have same base we

subtract the power

3- (b^m)^n = b^(mn) ⇒ if we have power over power we multiply

them

4- a^m × b^m = (ab)^m ⇒ if we multiply different bases with same

power then we multiply them ad put over the answer the power

5- b^(-m) = 1/(b^m) (for all nonzero real numbers b) ⇒ If we have

negative power we reciprocal the base to get positive power

6- If a^m = a^n , then m = n ⇒ equal bases get equal powers

7- If a^m = b^m , then a = b or m = 0

# Logarithmic rules:

1- $log_{a}b=n-----a^{n}=b$

2- $loga_{1}=0---log_{a}a=1---ln(e)=1$

3- $log_{a}q+log_{a}p=log_{a}qp$

4- $log_{a}q-log_{a}p=log_{a}\frac{q}{p}$

5- $log_{a}q^{n}=nlog_{a}q$

* Now lets solve the problems

# $3^{x+1}=9^{x+3}$

- Change the base 9 to 3²

∴ $9^{x+3}=3^{2(x+3)}=3^{2x+6}$

∴ $3^{x+1}=3^{2x+6}$

- Same bases have equal powers

∴ x + 1 = 2x + 6 ⇒ subtract x and 6 from both sides

∴ 1 - 6 = 2x - x

∴ -5 = x

* The solution x = -5

# ㏒(9x - 2) = ㏒(4x + 3)

- If ㏒(a) = ㏒(b), then a = b

∴ 9x - 2 = 4x + 3 ⇒ subtract 4x from both sides and add 2 to both sides

∴ 5x = 5 ⇒ divide both sides by 5

∴ x = 1

* The solution is x = 1

# $log_{6}(5x+4)=2$

- Use the 1st rule in the logarithmic equation

∴ 6² = 5x + 4

∴ 36 = 5x + 4 ⇒ subtract 4 from both sides

∴ 32 = 5x ⇒ divide both sides by 5

∴ 6.4 = x

* The solution is x = 6.4

# $log_{2}x+log_{2}(x-3)=2$

- Use the rule 3 in the logarithmic equation

∴ $log_{2}x(x-3)=2$

- Use the 1st rule in the logarithmic equation

∴ 2² = x(x - 3) ⇒ simplify

∴ 4 = x² - 3x ⇒ subtract 4 from both sides

∴ x² - 3x - 4 = 0 ⇒ factorize it into two brackets

∴ (x - 4)(x + 1) = 0 ⇒ equate each bract by 0

∴ x - 4 = 0 ⇒ add 4 to both sides

∴ x = 4

OR

∵ x + 1 = 0 ⇒ subtract 1 from both sides

∴ x = -1

- We will reject this answer because when we substitute the value

of x in the given equation we will find $log_{2}(-1)$ and this

value is undefined, there is no logarithm for negative number

* The solution is x = 4

# $log_{4}11.2=x$

- You can use the calculator directly to find x

∴ x = 1.7427

* The solution is 1.7427

# $2e^{8x}=9.2$ ⇒ divide the both sides by 2

∴ $e^{8x}=4.6$

- Insert ln for both sides

∴ $lne^{8x}=ln(4.6)$

- Use the rule $ln(e^{n})=nln(e)$ ⇒ ln(e) = 1

∴ 8x = ln(4.6) ⇒ divide both sides by 8

∴ x = ln(4.6)/8 = 0.190757

* The solution is 0.190757

7 0

3 years ago

Read 2 more answers

QUESTION 4 If 1 foot = 0.37 metres and 1 mile = 5,180 feet, find the number of kilometres in 15 miles. show how to work out the

Andreas93 [3]

Answer:

28.749 km

Step-by-step explanation:

in reality

1 foot = 0.3048 meters not 0.37

1 mile = 5,280 feet not 5,180

15 miles = 24.14016 kilometers

---------------------------

Using the numbers stated in the problem

0.37m/ft * 5180ft/mile * 15mile * 1km/1000m = 28.749 km

4 0

3 years ago

Need help please on geometry

zimovet [89]

To solve this we are going to use the formula for the area of a circle: $A= \frac{1}{4} \pi d^2$
where
$A$ is the area of the circle
$d$ is the diameter of the circle

We know from our problem that the area of our circle is 10 inches long, so $d=10in$ . Lets replace that value in our formula to find $A$ :
$A=\frac{1}{4} \pi (10in)^2$
$A=78.54in^2$

We can conclude that the correct answer is: 78.54 square inches, or in centimeters: 506.7 square cm.

4 0

3 years ago

Read 2 more answers

Please explain the problem to me step by step

JulijaS [17]

<h3>Solution:</h3>

$\frac{4y}{9} - \frac{2y}{3} = 10 \\ = > \frac{4y - 6y}{9} = 10 \\ = > \frac{ - 2y}{9} = 10 \\ = > y = 10 \times \frac{9}{ - 2} \\ = > y = - 5 \times 9 \\ = > y = - 45$

<h3>Answer:</h3>

-45

5 0

3 years ago

A medical device company knows that the percentage of patients experiencing injection-site reactions with the current needle is

mina [271]

Answer:

0.4970

Step-by-step explanation:

I might be wrong

6 0

3 years ago