To solve this logarithmic equation, we need to use the properties of logarithms to condense the terms and then isolate the variable.
Given equation:
log2^2 (x+4) + 4log2 (2x) - 9 = 0
First, apply the power property of logarithms, loga^n = nloga:
2(log2 (x+4)) + log2 (16x) - 9 = 0
Next, we can condense the terms by combining the logarithms with the same base:
log2 ((x+4)^2) + log2 (16x^4) - 9 = 0
Now, we can combine the logarithms into a single logarithm using the product property of logarithms, loga (x) + loga (y) = loga (xy):
log2((x+4)^2 * 16x^4) - 9 = 0
log2(16(x+4)^2*x^4) - 9 = 0
Now, convert the equation to exponential form:
2^(log2(16(x+4)^2*x^4)) = 2^9
16(x+4)^2*x^4 = 512
16(x^2 + 8x + 16)*x^4 = 512
16x^6 + 128x^5 + 256x^4 = 512
Divide the equation by 16:
x^6 + 8x^5 + 16x^4 = 32
Now, we need to solve this polynomial equation for x. This can be a bit complicated and may require numerical methods or factoring techniques.
To solve this logarithmic equation, we need to use the properties of logarithms to condense the terms and then isolate the variable.
Given equation:
log2^2 (x+4) + 4log2 (2x) - 9 = 0
First, apply the power property of logarithms, loga^n = nloga:
2(log2 (x+4)) + log2 (16x) - 9 = 0
Next, we can condense the terms by combining the logarithms with the same base:
log2 ((x+4)^2) + log2 (16x^4) - 9 = 0
Now, we can combine the logarithms into a single logarithm using the product property of logarithms, loga (x) + loga (y) = loga (xy):
log2((x+4)^2 * 16x^4) - 9 = 0
log2(16(x+4)^2*x^4) - 9 = 0
Now, convert the equation to exponential form:
2^(log2(16(x+4)^2*x^4)) = 2^9
16(x+4)^2*x^4 = 512
16(x^2 + 8x + 16)*x^4 = 512
16x^6 + 128x^5 + 256x^4 = 512
Divide the equation by 16:
x^6 + 8x^5 + 16x^4 = 32
Now, we need to solve this polynomial equation for x. This can be a bit complicated and may require numerical methods or factoring techniques.