Find the value of log base two of
10 plus log base two of 16 minus log base two of five without using a
Firstly, we’re going to recall the
order of operations. That tells us that when there is an
addition and a subtraction in the same sum, we simply move from left to right. So we’ll begin by evaluating log
base two of 10 plus log base two of 16. And then we’ll subtract log base
two of five. And so we’re going to recall some
laws of logarithms.
The first is sometimes called the
product law. And this says for a fixed base 𝑏
which is greater than zero and not equal to one and positive numbers 𝑥 one and 𝑥
two, log base 𝑏 of 𝑥 one times 𝑥 two is log base 𝑏 of 𝑥 one plus log base 𝑏 of
𝑥 two. Of course, the converse is
true. So we can say that to add
logarithms whose base is the same, we simply multiply the argument.
Similarly, with quotients, log base
𝑏 of 𝑥 one divided by 𝑥 two is log base 𝑏 of 𝑥 one minus log base 𝑏 of 𝑥
two. And so we can say that log base two
of 10 plus log base two of 16 is equal to log base two of 10 times 16. Usually, we’d look to evaluate
that, but we’re not going to just yet. Instead, we’re going to move
straight on to subtracting log base two of five. And we know this means we need to
divide the arguments. So we get log base two of 10 times
16 divided by five.
And we now see that since we didn’t
simplify, we can divide both the numerator and denominator of our fraction by five,
giving us log base two of two times 16 over one, which is log base two of 32. Remember, we’re trying to find the
value of this logarithm. So we recall the definition of a
logarithm. We say that log base 𝑏 of 𝑦
equals 𝑥 is equivalent to saying 𝑏 to the power of 𝑥 equals 𝑦. Well, here, our base is two. So we’re essentially asking what
exponent of two will give us 32. Well, we know that two to the fifth
power is 32, and so log base two of 32 must be five. The value of log base two of 10
plus log base two of 16 minus log base two of five is five.