Question Video: Finding the Solution Set of Logarithmic Equations Mathematics

Video Transcript

Given that 10 log 𝑥 plus 12 log 𝑦 minus log 𝑥 cubed 𝑦 to the fifth power is equal to seven multiplied by log seven plus log 10, determine the value of 𝑥.

In order to answer this question, we need to recall three of our laws of logarithms. Firstly, log 𝑎 plus log 𝑏 is equal to log 𝑎𝑏. Secondly, log 𝑎 minus log 𝑏 is equal to log of 𝑎 divided by 𝑏. Finally, 𝑛 log 𝑎 is equal to log 𝑎 to the 𝑛th power. Using the third law, we can rewrite 10 log 𝑥 as log 𝑥 to the power of 10 and 12 log 𝑦 as log 𝑦 to the power of 12.

The left-hand side becomes the sum of these two terms minus log of 𝑥 cubed multiplied by 𝑦 to the fifth power. On the right-hand side, we can simplify log seven plus log 10 by multiplying seven and 10. This gives us seven log 70. We can then use the first and second laws to rewrite the left-hand side as log of 𝑥 to the power of 10 multiplied by 𝑦 to the power of 12 divided by 𝑥 cubed multiplied by 𝑦 to the power of five.

Using the third law on the right-hand side, this is equal to log of 70 to the power of seven. Using our laws of exponents or indices, the expression inside the parentheses simplifies to 𝑥 to the power of seven multiplied by 𝑦 to the power of seven. This means that log of 𝑥 to the power of seven 𝑦 to the power of seven is equal to log of 70 to the power of seven.

If log 𝑎 is equal to log 𝑏, then the value of 𝑎 must be equal to the value of 𝑏. This means that in our question 𝑥 to the power of seven 𝑦 to the power of seven, which can be rewritten as 𝑥𝑦 to the power of seven, is equal to 70 to the power of seven. We can then take the seventh root of both sides of this equation, such that 𝑥 𝑦 is equal to 70. Dividing both sides of this equation by 𝑦 gives us 𝑥 is equal to 70 over 𝑦. This is the value of 𝑥 such that 10 log 𝑥 plus 12 log 𝑦 minus log of 𝑥 cubed 𝑦 to the power of five is equal to seven multiplied by log seven plus log 10.