Find the value of log base two of the log of 𝑥 to the 32nd power minus log base two of log of 𝑥 to the fourth power.
To simplify this logarithmic expression, we’re going to have to remember some log rules. First, we copy down our expression. And it’s probably most helpful to start inside the parentheses. We know that a log base 𝑏 of 𝑥 to the 𝑛 power is equal to 𝑛 times the log base 𝑏 of 𝑥. And that means inside the parentheses, we can rewrite log of 𝑥 to the 32nd power as 32 times the log of 𝑥. And we can rewrite log of 𝑥 to the fourth power as four times the log of 𝑥.
From there, we can bring down the rest of the expression. When we have log base 𝑏 of 𝑥 minus log base 𝑏 of 𝑦, we can simplify that by saying log base 𝑏 of 𝑥 over 𝑦. Notice here that we have the log base two of 32 times the log of 𝑥. And we’re subtracting the log base two of four times the log of 𝑥. The log from the first term becomes the numerator, and the log from the second term becomes the denominator. Here, our regular division rules apply. Log of 𝑥 over log of 𝑥 would be equal to one. And 32 over four would equal eight.
We now have a log base two of eight. When we’re working with log base two, we often want to find what’s inside the parentheses as an exponent with a base of two. We know that eight equals two cubed. And that means the log base two of eight is equal to log base two of two cubed. And we can use this rule from earlier to rewrite this as three times the log base two of two. We remember that log base 𝑏 of 𝑏 equals one. Therefore, log base two of two equals one. And this expression is equal to three.