Calculate the log

Question

Find the value (s) of x if 8(log₂ x - log_x 8) = 16

Answer 1

This is a difficult one for sure!

8(log₂x - log_x8) = 16

Solve for x

1.. First, let's divide both sides of the equation by 8 to isolate each expression.

-->log₂x - log_x8 = 2

2... Now we need to use some of the properties of logarithms to make the equation solvable.

We need to get rid of that log with a base of x.

We can use the change of base formula: log_ax = log_bx / log_ba

where our new base, b, is whatever number we choose. For this problem, let's make our new base 2, so it is the same as the first term in the equation:

log_x8 = log₂8 / log₂x

now we can replace log_x8 with our new term, and our equation becomes;

log₂x - (log₂8 / log₂x) = 2

log₂8=3, so our equation then becomes:

log₂x - (3 / log₂x) = 2

3.. Now that we have two of the same log term, we can replace them both with a variable to simplify the equation. Let log₂x = p and we have;

p - (3/p) = 2

solve for p:

subtract p from both sides

-3/p = 2 - p

multiply equation by p

-3 = 2p - p²

add three to both sides

0 = 2p - p² + 3

multiply by -1 to get a more typical quadratic formula, and order terms

p² - 2p + 3 = 0

solve for p by factoring

(p - 3) (p + 1) = 0

p= 3 or -1

4.. Now we know our term log₂(x) must be equal to either 3 or -1 for the expression to be valid.

so, we can set log₂(x) equal to each of these numbers to find the valid x values

log₂(x) = 3

using log rule b^c=a ~ log_ba=c ;

2³= x

x = 8

log₂(x) = -1

2^-1 = x

x = 1/2

So our answers are x= 8, 1/2.

*checking our work:

x=8

8(log₂(8) - log₈(8)) = 16

8(3 - 1) = 16

8 * 2 = 16

16 = 16

x=1/2

8(log₂(1/2) - log_1/2(8)) =16

8( (-1) - (-3) ) = 16

8(2) = 16

16 = 16

Answer 2

8(log₂(x) - log_x(8)) = 16 [Divide both sides by 8 to get:

(log₂(x) - log_x(8)) = 2 [Use the change of base equation to convert log base x to log base 2:

log₂(x) - (log₂(8)/log₂(x)) = 2 [Since log₂(8) = 3, simplify:

log₂(x) - 3/log₂(x) = 2 [multiply both sides of the equation by log₂(x) to get:

(log₂(x))² - 3 = 2log₂(x) [subtract 2log2(x) from both sides to get:

(log₂(x))² - 2log₂(x) - 3 = 0 [let w = log₂(x) and substitute to get:

w² - 2w - 3 = 0 [Factor:

(w - 3)(w + 1) = 0

w = 3 and w = -1

Now, back substitute w = log₂(x) to get:

1) log₂(x) = 3 which means 2³ = x or x = 8

and

2) log₂(x) = -1 which means 2^-1 = x or x = 1/2

Answer 3

Hi Sofea!

8(log₂ x - log_x 8) = 16 First divide by 8 on both sides.

(log₂ x - log_x 8) = 2 You can notice that 8 is a multiple of 2 and rewrite it as 2³.

(log₂ x - log_x 2³) = 2 Then using the log identity log_a x^b= b*log_ax , you can rewrite it.

(log₂ x - 3log_x 2) = 2 Afterward you can use the log identity log_ax=1/(log_xa)

(log₂ x - 3/(log₂ x) = 2 Then, you can substitute log₂ x with a variable such as y.

y-(3/y)=2 This can then be turned into a quadratic formula by multiplying everything by y.

y²-3=2y Bring everything to one side.

y²-2y-3=0 Find the factors.

(y-3)(y+1) Therefore, y=3 and y=-1. Remember that y=log₂ x

log₂ x = 3 To find x do 2^log2x=2³. You would be left with x=2³

x=8 Then, do the same with the -1 value to get x=2^-1

log₂ x = -1

x=1/2

The values of x would be 8 and 1/2. I hope that helped!