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Video: Computing Logarithmic Expressions Using Laws of Logarithms

Alex Cutbill

Find the value of log₄ 1280 − 2 log₄ 2 − log₄ 5 without using a calculator.

03:25

Video Transcript

Find the value of log base four of 1280 minus two times log base four of two minus log base four of five without using a calculator.

So we just write out the expression again. To find the value of this expression as we are asked to do, we’re going to repeatedly simplify it. The first thing we can notice is that we can simplify this term: two times a log base four of two. We do this using the law of logarithms that 𝑛 times log base 𝑏 of 𝑎 is equal to log base 𝑏 of 𝑎 to the power of 𝑛. And so two times log base four of two becomes log base four of two to the power of two.

Essentially what we’ve done is we’ve moved the multiplicative constant, two, from the front of a log base four of two to become the power of the two inside the log. And of course two to the power of two, or two squared, is just four. And so we have log base four of 1280 minus log base four of four minus log base four of five. Now all three terms are just the logarithm or log base four of a number, and we can simplify this by using another law of logarithms.

In general, log base 𝑏 of 𝑚 minus log base 𝑏 of 𝑛 is equal to a log base 𝑏 of 𝑚 over 𝑛. Applying this we see that log base four of 1280 minus log base four of four is log base four of 1280 over four. And of course, 1280 over four is just 320. And so our expression becomes log base four of 320 minus log base four of five.

So now we have another difference of our logarithms to the same base; log base four of 320 minus log base four of five. And you might like to think what the next line of working will therefore be. The rule we use is log base 𝑏 of 𝑚 minus log base 𝑏 of 𝑛 is log base 𝑏 of 𝑚 over 𝑛. And so we get log base four of 320 over five. And of course that is log base four of 64.

Are we done? Well, not quite. We can simplify further as it happens. Let 𝑥 equal log base four of 64. The exponential form of this relationship is four to the power of 𝑥 equals 64. Can you see what 𝑥 should be? Well, we know that four to the power of three, or four cubed, is 64. And so 𝑥 is equal to three.

So we write that our answer is three. It might be surprising that we get such a nice answer from such a complicated expression. Two times log base four of two is one. But log base four of 1280 and log base four of five are both irrational numbers. And so something really nice had to happen for us to get an answer of three at the end.