Simplify log base 𝑥 of the sixth root of 𝑥 to the fourth power.
Before we do any simplification with the log, let’s think about simplifying this exponent. We know that the 𝑛th root of 𝑥 is equal to 𝑥 to the one over 𝑛 power, which means we can rewrite this to say log base 𝑥 of 𝑥 to the fourth power to the one-sixth power. In this case, we’re taking a power of a power. And we can simplify 𝑥 to the 𝑎 power to the 𝑏 power by saying 𝑥 to the 𝑎 times 𝑏 power so that we have log base 𝑥 of 𝑥 to the four-sixths power. And this is where we’ll need a log rule.
If we have log of 𝑥 to the 𝑎 power, it will be equal to 𝑎 times the log of 𝑥. This means we now have four-sixths times log base 𝑥 of 𝑥. But what is this saying, log base 𝑥 of 𝑥? It’s asking us, 𝑥 to the what power would be equal to 𝑥? We know 𝑥 to the first power equals 𝑥. And this is where we get the rule that the log base 𝑏 of 𝑏 equals one. And that means we substitute one for log base 𝑥 of 𝑥, which gives us four-sixths times one.
Our instructions told us to simplify. And we haven’t simplified this expression completely because four and six are both divisible by two. If we divide the numerator and the denominator by two, we get two-thirds, which tells us that the log base 𝑥 of the sixth root of 𝑥 to the fourth power is two-thirds.