Question Video: Simplifying Logarithmic Expressions Using Laws of Logarithms Mathematics

Simplify log₃ 16 × log₂ 243.

02:10

Video Transcript

Simplify log 16 to the base three multiplied by log 243 to the base two.

As the bases of our two terms are different, in this case, three and two, our first step here is to rewrite 16 and 243 in index form. 16 is equal to two to the power of four as two multiplied by two multiplied by two multiplied by two is equal to 16. 243 is equal to three to the power of five. We can, therefore, rewrite the expression as log two to the power of four with a base of three multiplied by log of three to the power of five with a base of two.

One of our laws of logarithms states that log of 𝑥 to the power of 𝑛 is equal to 𝑛 multiplied by log 𝑥. Our expression then simplifies to four log two to the base three multiplied by five log three to the base two. At this point, we notice that our first term is log two to the base three and our second term is log three to the base two.

Another one of our laws of logarithms states that log of 𝑦 to the base 𝑥 multiplied by log of 𝑥 to the base 𝑦 is equal to one. This is because log of 𝑥 to the base 𝑦 is the reciprocal of log 𝑦 to the base 𝑥. Our expression can, therefore, simplify to four multiplied by five as log two to the base three multiplied by log three to the base two is equal to one. Four multiplied by five is equal to 20.

This means that log 16 to the base three multiplied by log 243 to the base two is equal to 20.

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