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Question Video: Computing the Perimeter of a Triangle with Logarithmic Side Lengths Mathematics • 10th Grade

Determine the perimeter of the figure.

02:27

Video Transcript

Determine the perimeter of the figure below.

In this question, we’re given a triangle with two of the sides shown. We need to determine the perimeter of this triangle. To do this, let’s start by recalling what we mean by the perimeter of a shape. We recall the perimeter of a polygon is the sum of all of its side lengths. And in the figure, we’re given two of these side lengths. And in fact, we can see that this is an isosceles triangle. This is represented by the cut in the line. The length of 𝐴𝐶 must be equal to the length of 𝐴𝐵. Therefore, the length of 𝐴𝐶 is the log base four of eight centimeters.

And now the sum of these three side lengths will be the perimeter 𝑃 of the triangle. It’s the logarithm base four of eight plus the logarithm base four of eight plus the logarithm base four of four. And we can give this the unit of centimeters. However, we’ll do this at the end. There’s actually many different ways of evaluating this expression. We’ll only go through one of these. First, we recall for any positive real number 𝑏 not equal to one, the log base 𝑏 of 𝑏 is equal to one. Therefore, the logarithm base four of four is equal to one. Next, our other two terms are equal, so we can add these together to get two times the log base four of eight.

And since we can’t yet evaluate this expression, let’s simplify this further by using the power rule for logarithms. This tells us for any real number 𝑃, positive real numbers 𝑏 and 𝑥, where 𝑏 not equal to one, 𝑃 times the logarithm base 𝑏 of 𝑥 is equal to the log base 𝑏 of 𝑥 to the power of 𝑃. In other words, we can bring the factor of two inside of our logarithm as an exponent of eight. This gives us the log base four of eight squared, which is of course equal to 64. And this is a useful result because 64 is equal to four cubed. This then gives us the log base four of four cubed plus one.

And now there’s a few different ways we can evaluate this expression. We could just use the power rule for logarithms to take the exponent of three back outside of our logarithm and then use the log base 𝑏 of 𝑏 is equal to one. However, it’s easier to use the fact that the logarithm base four will be the inverse of the exponential function with base four. In other words, the log base 𝑏 of 𝑏 to the power of 𝑛 is just equal to 𝑛 for any positive real number 𝑏 not equal to one. So, the log base four of four cubed is three, giving us three plus one, which we can evaluate is equal to four. And finally, since the length we’re given in centimeters, we can add the unit of centimeters, which means we were able to show the perimeter of the triangle given to us in the question is four centimeters.

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