Please, help me solve this.
log[2]x * log[2](x/4) = 8
To solve the equation log[2]x * log[2](x/4) = 8, we can apply some properties of logarithms.
First, let's rewrite the equation using the multiplication rule of logarithms:
log[2]x + log[2](x/4) = 8
Next, let's simplify the logarithmic expressions using the quotient rule of logarithms:
log[2]x + [log[2]x - log[2]4] = 8
log[2]x + log[2]x - log[2]4 = 8
Now, let's combine the logarithmic terms:
2 * log[2]x - log[2]4 = 8
Next, let's simplify the logarithms by using the base change rule. We can change log[2] to log[10] to make the calculation easier:
2 * (log[10]x / log[10]2) - (log[10]4 / log[10]2) = 8
Simplifying further:
2 * log[10]x / log[10]2 - log[10]4 / log[10]2 = 8
Now, let's apply the properties of logarithms to combine the terms:
[2 * log[10]x - log[10]4] / log[10]2 = 8
Next, let's simplify the numerator:
[log[10]x^2 - log[10]4] / log[10]2 = 8
Using the properties of logarithms again, we can simplify further:
log[10](x^2 / 4) / log[10]2 = 8
Now, let's cancel out the logarithms by multiplying both sides by log[10]2:
log[10](x^2 / 4) = 8 * log[10]2
Now, we have a logarithmic equation in base 10. We need to solve for x, so let's rewrite the equation in exponential form:
x^2 / 4 = 2^8
Simplifying further:
x^2 / 4 = 256
Multiplying both sides by 4:
x^2 = 1024
Taking the square root of both sides:
x = ±32
Therefore, the solutions to the equation log[2]x * log[2](x/4) = 8 are x = 32 and x = -32.