Question Video: Using Laws of Logarithms to Verify Equality Statements Mathematics

Is it true that log_(π‘Ž) (π‘₯ + 𝑦) = log_(π‘Ž) π‘₯ + log_(π‘Ž) 𝑦?

02:52

Video Transcript

Is it true that the logarithm base π‘Ž of π‘₯ plus 𝑦 is equal to the logarithm base π‘Ž of π‘₯ plus the logarithm base π‘Ž of 𝑦?

In this question, we’re given a logarithmic equation involving values of π‘Ž, π‘₯, and 𝑦, and we need to determine if this equation is true. We might be tempted to try and prove this is true by using the definition of a logarithm. However, since this isn’t one of our given rules of logarithms, it’s usually easier to check whether the statement is true for a few values first. So we want to choose some values of π‘Ž, π‘₯, and 𝑦 to check if both sides of the equations are equal.

To do this, let’s recall some laws of logarithms to make both sides of the equation easy to evaluate. First, we recall for any base 𝑏 that’s a positive real number not equal to one, the log base 𝑏 of 𝑏 is equal to one. Therefore, if we chose π‘Ž is equal to three, π‘₯ is equal to three, and 𝑦 is equal to three, both terms on the right-hand side of this equation would be of the form log base 𝑏 of 𝑏. So we could evaluate both of these terms.

For the statement to be true, it would need to hold for all values of π‘Ž, π‘₯, and 𝑦 which leave the equation well defined. So let’s check if this is true for π‘Ž is three, π‘₯ is three, and 𝑦 is three. The right-hand side of our equation becomes log base three of three plus log base three of three, which is equal to one plus one, which of course is just equal to two. Let’s now check if the same is true on the left-hand side of the equation. When π‘Ž is three, π‘₯ is three, and 𝑦 is three, the left-hand side of the equation becomes the log base three of three plus three, which we can simplify to give us the log base three of six. For the statement to be true, this value needs to be equal to two.

However, we can show that this is not true. For example, for any positive real number 𝑏, we know the log base 𝑏 of 𝑏 to the 𝑛th power is equal to 𝑛, since the logarithmic function of base 𝑏 is the inverse function of 𝑏 to the 𝑛th power. An application of this result tells us that the log base three of three squared is equal to two. And of course three squared is equal to nine. It’s not equal to six.

Therefore, both sides of the equation are not equal for these values of π‘Ž, π‘₯, and 𝑦, so the statement is false. However, for extra clarity, let’s also use our calculators to evaluate the log base three of six. To two decimal places it’s 1.63. Once again, this is not equal to the right-hand side of the equation. So we can say that no, it is not true that the log base π‘Ž of π‘₯ plus 𝑦 is equal to the log base π‘Ž of π‘₯ plus the log base π‘Ž of 𝑦.

However, there is a statement that’s very close to this which is true. It’s called the product rule for logarithms. It states for any positive real numbers π‘Ž, π‘₯, and 𝑦, where π‘Ž is not equal to one, the log base π‘Ž of π‘₯ times 𝑦 is equal to the log base π‘Ž of π‘₯ plus the log base π‘Ž of 𝑦, with the big difference being we’re taking the product of π‘₯ and 𝑦 inside of our logarithm. We’re not taking the sum. In either case, we can show that no, it is not true that the log base π‘Ž of π‘₯ plus 𝑦 is equal to the log base π‘Ž of π‘₯ plus the log base π‘Ž of 𝑦.

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