log4x^4 + log4^2 = 3

Solve the logarithmic equation.

log4x^4 = log4 + 4logx

log4^2 = 2log4

log4 + 4logx + 2log4 = 3
3log4 + 4logx = 3
4logx = 3(1-log4)

Now, if we're dealing with base 10, 1 = log10, so 1-log4 = log(10/4) = log(5/2)

4 logx = 3log(5/2)
logx = 3/4 log(5/2)

x = 10^(3/4) * l0^(log 5/2)
x = 10^3/4 * 5/2
x = 5/2 10^(3/4)
x = 5/2 ∜1000

To solve the logarithmic equation log4x^4 + log4^2 = 3, we can use the properties of logarithms.

Let's start by simplifying the equation. According to the logarithmic property, log a + log b = log (a * b). Applying this property to the given equation, we have:

log4x^4 + log4^2 = log4(x^4 * 4^2)

Simplifying further:

log4x^4 + log4^2 = log4(x^4 * 16)

Next, we can use another logarithmic property, log a^n = n * log a, which states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. Applying this property to the equation, we have:

4log4x + 2 = log4(16x^4)

Now we can use one more logarithmic property, log a^b = b * log a, to simplify the equation further:

4log4x + 2 = 4log4(4x)

Now that the bases of the logarithms are the same, we can equate the exponents:

4log4x + 2 = 4(1 + log4x)

Simplifying the right side:

4log4x + 2 = 4 + 4log4x

Now let's consolidate the terms with log4x on one side of the equation:

4log4x - 4log4x = 4 - 2

0 = 2

Here we see that 0 does not equal 2, which means there is no solution to this equation. Hence, the given equation log4x^4 + log4^2 = 3 has no solution.