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VARVARA [1.3K]
3 years ago
11

Solve the following logarithmic equations. ln(x^6) = 36

Mathematics
1 answer:
Hoochie [10]3 years ago
8 0

Answer:

The solution is x = e⁶

Step-by-step explanation:

Hi there!

First, let´s write the equation

ln(x⁶) = 36

Apply logarithm property: ln(xᵃ) = a ln(x)

6 ln(x) = 36

Divide both sides of the equation by 6

ln(x) = 6

Apply e to both sides

e^(ln(x)) = e⁶

x = e⁶

The solution is x = e⁶

Let´s prove why e^(ln(x)) = x

Let´s consider this function:

y = e^(ln(x))

Apply ln to both sides of the equation

ln(y) = ln(e^(ln(x)))

Apply logarithm property: ln(xᵃ) = a ln(x)

ln(y) = ln(x) · ln(e)         (ln(e) = 1)

ln(y) = ln(x)

Apply logarithm equality rule: if ln(a) = ln(b) then, a = b

y = x

Since y = e^(ln(x)), then x =e^(ln(x))

Have a nice day!

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3 years ago
Dado el número N = 4752a, averigua qué valor ha de tomar "a" para que: a) N sea divisible por tres. b) N sea divisible por cinco
lara [203]

Answer:

I will answer it in English:

a) We have te number N = 4752a

if we want to divide it by 3, then the addition of all the digits must be a multiple of 3.

Sum = 4 + 7 + 5 + 2 + a = 6 + 7 + 5 + a = 18 + a

and 18 is a multiple of 3, then we can have, a = 0, a = 3, a = 6, a= 9

b) A number is divisible by 5 if it ends on 5 or 0, so the possible values of a are 0 and 5.

c) we know that 15*3 = 45

then 15*3000= 45000

now we have:

4752a - 45000 = 252a

15*13 = 2520

then we have that a must be equal to zero, and:

15*(13 + 3000) =  47520

d) We want a number that when we divide it by 7, the surpass is 5.

a number is divisible by 7 if the difference between the number without the units and the double of the units is 0 or a multiple of 7.

in N = 4752x the unit is a, then we have:

D = 4752 - 2*x

now, let's choose a such we have a multiple of 7

we know that 5746 is a multiple of 7, so we can choose x= 3 and get:

D = 4752 - 2*3 =   5746  

So the number N = 47523 is a multiple of 7, and when we divide it by 7 we will have a surpass of 0.

Then when we divide N + 5 by 7, we will have a surpass of 7.

This is: N + 5 = 47528

This means that the actual value of a that we need is 8.

5 0
3 years ago
A sign language club made $627. 50 from selling popcorn at a festival. The popcorn cost the club $95. 0. Which expression repres
solmaris [256]

The expression representing the amount of money that was raised for each member of the club is \dfrac{627.50 - 95.0}{m} (in dollars)

<h3>How to interpret the division? </h3>

When 'a' is divided by 'b', then the result we get from the division is the part of 'a' that each one of 'b' items will get.

Thus, if 10 mangoes are there, and 2 people, then 10 ÷ 2 is the number of mangoes each person would get, which is 5.

Division, thus, can be interpreted as equally dividing the number that is being divided in total x parts, where x is the number of parts the given number is divided.

Thus, a \div b = a divided in b equal parts.

Also, we can write: a \div b = a \times \dfrac{1}{b}

(it is since a = a \times 1 so a/b = 1 \times (a/b) = (1/b) \times a )

<h3>How to find the profit earned?</h3>

Profit earned is the extra money which we make after deducing the cost price of the process we're doing.

So, we get:

Profit earned = Total amount earned - Amount spent initially for preparation to earn

P = Selling price (S.P.) = Cost price (C.P.)

For the given case, we're given that:

  • Amount made by the sign language club by selling popcorn at a festival = $627.50
  • The cost price for the popcorn to the club = $95.0

The money raised = Amount made by selling popcorn - amount costed to the club for getting popcorn

Money raised = $627.50 - $95.0

If it is given that: m = number of members in the considered club, then:

Money each member of the considered club will get = Money raised divided in equal m parts = \dfrac{627.50 - 95.0}{m} (in dollars)

(we don't write symbols like of currency generally, and understand it from context.)

Thus, the expression representing the amount of money that was raised for each member of the club is \dfrac{627.50 - 95.0}{m} (in dollars)

Learn more about division here:

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Answer: L = 1. K=2

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