Question Video: Computing Logarithms by Using Laws of Logarithms | Nagwa Question Video: Computing Logarithms by Using Laws of Logarithms | Nagwa

# Question Video: Computing Logarithms by Using Laws of Logarithms Mathematics • Second Year of Secondary School

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Which of the following is equal to (5 log 3)/(log 4 + log 6)? [A] log 3 [B] log₂₄ 15 [C] log₂₄ 243 [D] log 15 [E] log 243

02:05

### Video Transcript

Which of the following is equal to five log three over log four plus log six?

This expression might look a little bit strange as our logs seem to have no base. If a log has no base, we generally assume that the base is equal to 10. And so we rewrite our fraction as five log base 10 of three over log base 10 of four plus log base 10 of six.

So we’re next going to recall some rules for logarithms. Firstly, we know that log base 𝑏 of 𝑥 one plus log base 𝑏 of 𝑥 two is log base 𝑏 of 𝑥 one times 𝑥 two. As long as our bases are the same, we simply multiply the arguments. And so the denominator of our fraction is going to become log base 10 of four times six, which is log base 10 of 24.

And what about our numerator? Well, log base 𝑏 of 𝑥 to the power of 𝑝 for real constants 𝑝 is the same as 𝑝 times log base 𝑏 of 𝑥. The converse is true. So we can write our denominator as log base 10 of three to the fifth power. But three to the fifth power is 243. And so we have log base 10 of 243 over log base 10 of 24.

Note that we have a fraction with two logarithms whose bases are equal. And so we can use the change of base formula. This says that log base 𝑏 of 𝑥 one divided by log base 𝑏 of 𝑥 two can be written as log base 𝑥 two of 𝑥 one. So, essentially, if the bases are the same, we make the argument of our denominator the new base. And the argument of our numerator becomes the new argument. In this case then, the base of our logarithm is 24 and its new argument is 243. So we can write our fraction as log base 24 of 243. And our correct answer is therefore (C).

Note that this actually means it didn’t matter what base we assumed. Because we ended up with two logarithms with the same base, we simply apply the change of base formula. We could’ve chosen base two or base three. But remember, the general convention is to assume that it’s base 10.

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