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2 years ago

how to solve for 8^x=2 (I know the answer is 1/3 but i have to show work and i don't know the process to solve it)

Mathematics

1 answer:

Lunna [17]2 years ago

5 0

Answer:

x = 1/3 . . . . math facts or logarithms are involved; take your pick

Step-by-step explanation:

When you are solving for a variable that is in an exponent, logarithms are often useful. Taking the log of both sides of this equation, you have ...

log(8^x) = log(2)

Using the rules of logarithms, that is ...

x·log(8) = log(2)

x = log(2)/log(8) . . . . . divide by the coefficient of x

You can find the value of this on your calculator, and it will tell you the value is 0.333333333333 or as many digits as your calculator displays. That is a clue that the exact answer is probably 1/3.

You should recognize that 8 = 2·2·2 = 2^3, so log(8) = 3log(2) and the above solution becomes ...

x = log(2)/(3log(2)) = 1/3

Recognizing that 8 = 2^3, you can make that substitution into the original equation to get ...

(2^3)^x = 2

2^(3x) = 2^1

3x = 1 . . . . . . . matching exponents; equivalent to taking logs base 2

x = 1/3 . . . . . . divide by 3

All of the above using 2 as a base of exponents is just dancing around the fact that you already know the math fact ...

8^x = 2 = 8^(1/3)

x = 1/3 . . . . . equating exponents; equivalent to taking logs base 8

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2 years ago

Sally is having a problem with her puppy leaving the yard so she decides to build a new fence. The length of the yard is 8 feet

Rufina [12.5K]

Please check the attached file for the diagram of Sally's yard.

Let $w$ be the width of the yard.

Let $l$ be the length.

We are given that the length of the yard is 8 feet more than 4 times the width. Let us convert this into a mathematical equation.

4 times the width will be $4w$ . Now, 8 feet, or 8 more than 4 times the width will be $4w+8$ . We know that this is the value of the length of the yard and thus we can write this as:

$l=4w+8$

As we can see we have found the relationship between the length, $l$ and the width, $w$ of the yard.

We have been told that the total length of the fence required for the yard is 106 feet. This obviously means that the perimeter of the rectangular yard is 106 feet.

Now, we know that the perimeter of a rectangle is given by:

Perimeter= $2(l+w)=2((4w+8)+w)$ this is because we know that the length, $l$ is given by $l=4w+8$ .

Continuing further we get:

$106=8w+16+2w=10w+16$ This is because Perimeter=length of the fence=106 (given)

Therefore, $10w=90$

or, $w=9$ so, we just now found the width of the yard which is 9 feet. We are required to find the length. We can easily find the length using the relationship between $w$ and $l$ which is: