Find the value of log of one over 32 to the base four plus 14 log two to the base four plus log 16 to the base four minus log of a half to the base four minus log 64 to the base four without using a calculator.
As the base of all five of our terms is the same, we can use our laws of logarithms. Before doing this, we notice that one over 32, 16, one-half, and 64 can all be written as two to some power or exponent. 16 is equal to two to the power of four. 64 is equal to two to the power of six. One-half is equal to two to the power of negative one, as any number raised to the power of negative one is the reciprocal of that number. The reciprocal of two is one-half. This means as two to the power of five is equal to 32, two to the power of negative five will be equal to one over 32.
We can therefore rewrite our initial expression as shown. Log of two to the power of negative five plus 14 log two plus log two to the power of four minus log two to the power of negative one minus log two to the power of six. All of these terms are to the base four.
One of our laws of logarithms states that log 𝑎 to the power of 𝑛 is equal to 𝑛 multiplied by log 𝑎. We can bring our four powers or exponents down in front of each term. This gives us negative five log two plus 14 log two plus four log two plus log two minus six log two. Negative five plus 14 plus four plus one minus six is equal to eight. Our expression simplifies to eight log two to the base four.
Let’s now consider how we can calculate the value of log two to the base four. We know that if log 𝑏 to the base 𝑎 is equal to 𝑐, then 𝑎 to the power of 𝑐 is equal to 𝑏. In this question, 𝑎 is equal to four, 𝑏 is equal to two, and we are trying to calculate the value of 𝑐. Four to the power of 𝑐 is equal to two. As the square root of four is equal to two, then four to the power of a half equals two. Our value of 𝑐 is one-half, so log two to the base four is one-half.
To calculate the value of our expression, we need to multiply eight by one-half. This is equal to four, which we could check using a calculator.