Question Video: Simplifying Rational Expressions Using Laws of Logarithms Mathematics

Video Transcript

Simplify one divided by log base two of 12 plus one divided by log base eight of 12 plus one divided by log base nine of 12.

In this question, we’re asked to simplify an expression involving logarithms. And there’s a few different ways we could go about doing this. For example, because each of the three terms is a rational expression, we could add these three terms together. However, this would give quite a complicated expression. Instead, let’s try and simplify each term individually. To do this, we note that each term is one divided by some logarithmic expression. And we recall the change of base formula for logarithms allows us to simplify the quotient to two logarithms. And this tells us for positive real numbers 𝑏, 𝑥 one, and 𝑥 two, where 𝑏 and 𝑥 two are not equal to one, the logarithm base 𝑏 of 𝑥 one divided by the logarithm base 𝑏 of 𝑥 two is equal to the logarithm base 𝑥 two of 𝑥 one.

However, we can’t yet apply this to the three terms given to us in the question because we need to rewrite the numerator as a logarithm. And the base of this logarithm will need to be the same as the base of the logarithm in the denominator. The base needs to be two. And we can do this by recalling the logarithm base 𝑏 of 𝑏 will be equal to one for any positive real value of 𝑏 not equal to one. And now we can use these two results to simplify all three terms of this expression.

We’ll start with the first term. One divided by log base two of 12 is equal to log base two of two divided by log base two of 12. And now that we’re taking the quotient of two logarithms of the same base, we can use the change of base formula. This then gives us the logarithm base 12 of two. We can then apply the same process to our upper two terms. For the second term, we rewrite the numerator as the logarithm base eight of eight. And once again we’re taking the quotient of two logarithms to the same base. So we can apply the change of base formula. This then gives us that the second term of this expression is equal to the logarithm base 12 of eight.

Finally, we apply the same process to our third term. We rewrite the numerator as the logarithm base nine of nine and then use the change of base formula. This allows us to write this term as the logarithm base 12 of nine. We can now substitute these into the expression we’re given in the question. This gives us the logarithm base 12 of two plus the logarithm base 12 of eight plus the logarithm base 12 of nine. And we can now see this is the sum of three logarithmic expressions with the same base.

And we recall we can simplify logarithmic expressions of this form by using the product rule for logarithms. This tells us for any positive real numbers 𝑏, 𝑥, and 𝑦, where 𝑏 is not equal to one, the logarithm base 𝑏 of 𝑥 times 𝑦 is equal to the logarithm base 𝑏 of 𝑥 plus the logarithm base 𝑏 of 𝑦. On the right-hand side of this expression, we have the sum of logarithms of the same base. And this is the same as the expression we’re given. However, we can see we’re adding three terms of this form.

We could apply this result two separate times, once on the first two terms and then again on the third term. However, it’s still just going to be the logarithm base 𝑏 of the product. So instead, we can just apply this result all at once. It’s the logarithm base 12 of two multiplied by eight multiplied by nine. And we can simplify this. Two times eight times nine is equal to 144, which is of course 12 squared. Therefore, this expression simplifies to give us the log base 12 of 12 squared.

And we can evaluate this expression directly from the definition of a logarithm. Or alternatively, we can just use the fact the logarithm base 𝑏 of 𝑏 to the 𝑛th power is equal to 𝑛 for any positive real value of 𝑏 not equal to one. And this gives us our final answer. Therefore, we were able to show one divided by log base two of 12 plus one divided by log base eight of 12 plus one divided by log base nine of 12 is equal to two.