(6) Which equation is equivalent to

4^-x=8?

a. log8 4=x

b. logx = -4

c. log -x 4=8

d. log 4 8= -x

1/4^x = 8

4^x = 1/8
x log4 (4) = log4 (1/8)
but log4 (4) = 1 because b^logb(a) = a
x = log4(1/8) = log4(1) - log4(8)
but log4(1) = 0 for the same reason
so
x = -log4(8) or -x = log4 (8)

To find the equation that is equivalent to 4^-x=8, we need to rewrite the equation using logarithms.

The logarithm of a number is the power to which another fixed value, called the base, must be raised to produce that number. In this case, the base will be the number 4.

We can rewrite 4^-x=8 using logarithms as follows:

log4(4^-x) = log4(8)

Since the base of the logarithm is 4, we can simplify the left side using the logarithmic property: log_b(b^x) = x.

Therefore, the equation becomes:

-x = log4(8)

Now, to find which of the given options is equivalent to this equation, we can examine each option:

a. log8 4 = x: This equation has a different base (8 instead of 4). It is not equivalent.

b. logx = -4: This equation is not related to the original equation. It is not equivalent.

c. log -x 4 = 8: This equation has a negative base (-x) which is not allowed in logarithms. It is not equivalent.

d. log 4 8 = -x: This equation is equivalent to the original equation. By using the logarithmic property, we can rewrite it as -x = log4(8).

Therefore, the correct answer is d. log 4 8 = -x, which is equivalent to the equation 4^-x = 8.