expand log2 (8x^2+48X+72)
Log2(8x^2+48x+72).
Log2(8)(x^2+6x+9) =
Log2(8)(x+3)^2 =
Log2(8)+2Log2(x+3) =
3 + 2Log2(x+3).
Well, well, well, looks like someone wants to expand their logarithmic horizons! Let's give it a shot, shall we?
Now, we're dealing with the logarithm base 2 of the expression 8x^2 + 48x + 72. To expand this, we need to break it down into its logarithmic components.
First things first, let's rewrite the expression as a power of 2:
8x^2 + 48x + 72 can be factored into (8(x^2 + 6x + 9)), and if you remember your algebra, that's actually (2^3(x + 3)^2). Neat, huh?
Now, let's use the properties of logarithms to expand this bad boy. According to the power rule of logarithms, log base 2 of (a*b) is equal to log base 2 of a plus log base 2 of b. So, applying this rule to our expression, we get:
log base 2 of (2^3) + log base 2 of (x + 3)^2
Now, the logarithmic base 2 of 2^3 is simply 3. So we can simplify it further as:
3 + 2 * log base 2 of (x + 3)
And there you have it! We've expanded log base 2 of (8x^2 + 48x + 72) to 3 + 2 * log base 2 of (x + 3). Ta-da!
To expand the expression log2(8x^2+48x+72), we need to rewrite it using the properties of logarithms.
First, let's simplify the expression inside the logarithm:
8x^2 + 48x + 72 = 8(x^2 + 6x + 9)
Next, let's factor the quadratic inside the parentheses:
x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2
Now, we can rewrite the expression as:
log2(8(x + 3)^2)
Using the property of logarithms, we can bring down the exponent:
= log2(8) + log2((x + 3)^2)
Simplifying further, we know that log2(8) is equal to 3:
= 3 + log2((x + 3)^2)
Finally, the expanded form of log2(8x^2+48x+72) is 3 + log2((x + 3)^2).
To expand the expression log2(8x^2 + 48x + 72), we first need to use the properties of logarithms. In this case, we can use the property that allows us to bring the exponent down in front of the logarithm.
The expression log2(8x^2 + 48x + 72) can be rewritten as:
log2((8 * x^2 * 6) + (8 * x * 6) + (8 * 9))
Now, let's simplify it further:
log2(48x^2 + 48x + 72)
Now, we can factor out a common factor of 24 from each term:
log2(24 * (2x^2 + 2x + 3))
Next, we can rewrite 2x^2 + 2x + 3 as a perfect square trinomial:
2x^2 + 2x + 3 = (x^2 + x) + (x + 1) + 3 = (x(x+1)) + (x+1) + 3 = (x+1)(x+1) + 3 = (x+1)^2 + 3
Substituting back into our expression:
log2(24 * ((x+1)^2 + 3))
Now, we can use another property of logarithms, which allows us to separate the product inside the logarithm into a sum of logarithms:
log2(24) + log2((x+1)^2 + 3)
Finally, we have expanded the expression log2(8x^2 + 48x + 72) as:
log2(24) + log2((x+1)^2 + 3)