Solve for X 4^X=8X

Answer 1

To solve for x, we can begin by taking the logarithm of both sides of the equation:

log(4^x) = log(8x)

Using the logarithm rule log(a^b) = b*log(a), we can simplify the equation:

x*log(4) = log(8x)

Now, let's solve for x:

x = log(8x) / log(4)

To further simplify, we can use the change of base formula:

x = log(8x) / log(4)
x = ln(8x) / ln(4)

Let's multiply both sides by ln(4):

x * ln(4) = ln(8x)

We can then distribute ln(4) on the left side and rearrange:

x * ln(4) - ln(8x) = 0

Now, let's factor out x:

x (ln(4) - ln(8x)) = 0

From this equation, we see that either x = 0 or ln(4) - ln(8x) = 0.

For x = 0, we have:

0 * ln(4) - ln(8(0)) = 0

0 - ln(0) = 0

This is undefined since ln(0) is undefined.

Therefore, we need to solve for ln(4) - ln(8x) = 0:

ln(4) - ln(8x) = 0

Using the logarithm rule ln(a) - ln(b) = ln(a/b), we can simplify:

ln(4/8x) = 0

Now, we can convert this equation back to exponential form:

4/8x = e^0

Since e^0 equals 1, we have:

4/8x = 1

We can then multiply both sides by 8x:

4 = 8x

Finally, divide both sides by 8:

4/8 = x

x = 1/2

Therefore, the solution to the equation is x = 1/2.