Given that 𝐿 and 𝑀 are two roots of the equation 𝑥 squared minus 36𝑥 plus 128 equals zero, determine the value of log base two of 𝐿 plus log base two of 𝑀.
Our end goal here is to identify what log base two 𝐿 plus log base two 𝑀 is. However, before we can calculate the log base two of 𝐿 and 𝑀, we have to first find out what 𝐿 and 𝑀 are. And 𝐿 and 𝑀 are the roots of this equation, which means our first step here is to factor the equation 𝑥 squared minus 36𝑥 plus 128 to see if we can find 𝐿 and 𝑀, the roots. Since we’re dealing with 𝑥 squared, we know that both of our terms will be an 𝑥 plus or minus some constant. We’ll need to find two constants that when multiplied together equal 128, and when added together equal negative 36.
One strategy we can use here is a factor tree. We can try and factor 128. If we start with two, two times 64 equals 128. And then, we can say 64 is divisible by two. Two times 32 equals 64. Now, remember, we’re looking for values that multiply together to equal 128 and when added together equal negative 36. In this tree, we’ve recognized that 32 is a factor of 128. We can say that two times two times 32 equals 128, which means we can also say four times 32 equals 128. Not only that, but negative four times negative 32 equals 128. And negative four plus negative 32 equals negative 36. This means we’ve found our factors.
𝑥 minus four times 𝑥 minus 32 is set equal to zero, which means we need to calculate 𝑥 minus four equals zero and 𝑥 minus 32 equals zero. In this case, we have a root at 𝑥 equals four and a root at 𝑥 equals 32. We can then say that 𝐿 equals four and 𝑀 equals 32. It wouldn’t matter if we reverse these. We could say 𝑀 equals four and 𝐿 equals 32. Now, we’re ready to consider log base two of 𝐿 plus log base two of 𝑀, log base two of four plus log base two of 32.
Remember, log base two of four is asking the question, two to what power equals four? Since we know that two squared equals four, we can say that log base two of four equals two. And log base two of 32 is asking, two to what power equals 32? We know that two to the fifth power equals 32, which makes log base two of 32 equal to five. And when we combine those, we see that we get seven. Under these conditions, log base two of 𝐿 plus log base two of 𝑀 will equal seven.