Solve for X 4^X=8X
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.
To solve the equation 4^x = 8x, we can start by taking the logarithm base 4 of both sides:
log4(4^x) = log4(8x)
Using the logarithm property that logb(b^y) = y, we get:
x = log4(8x)
Now, let's simplify the equation further. We can rewrite 8 as 2^3:
x = log4(2^3x)
Next, using the logarithm property logb(a^c) = c * logb(a), we can bring down the exponent:
x = 3x * log4(2)
Using the base conversion formula logb(a) = logc(a) / logc(b), we can convert the base 4 logarithm to a base 2 logarithm:
x = 3x * log2(2) / log2(4)
Since log2(2) is equal to 1, and log2(4) is equal to 2, we can substitute these values into the equation:
x = 3x * 1 / 2
Simplifying further, we get:
x = 3x / 2
To solve for x, let's get rid of the fractions by multiplying both sides of the equation by 2:
2x = 3x
Next, subtract 2x from both sides:
0 = x
Therefore, the solution to the equation 4^x = 8x is x = 0.