Math

QuestionIdentify the TRUE statement among these options: A) log8+log2=log10\log 8+\log 2=\log 10, B) log5×log3=log8\log 5 \times \log 3=\log 8, C) log9log3=log3\log 9-\log 3=\log 3, D) log6log2=log4\log 6-\log 2=\log 4.

Studdy Solution

STEP 1

Assumptions1. We are dealing with logarithms of the same base (base10 in this case). . We are using the properties of logarithms, which include the product rule, quotient rule, and power rule.

STEP 2

Let's start by examining option A. According to the product rule of logarithms, the sum of the logarithms of two numbers is equal to the logarithm of their product. So, we can rewrite log8+log2\log8+\log2 as log(8×2)\log (8 \times2).

STEP 3

Calculate the product of8 and2.
8×2=168 \times2 =16So, log8+log2=log16\log8+\log2=\log16, not log10\log10. Therefore, option A is FALSE.

STEP 4

Next, let's examine option B. There's no rule that allows us to multiply two logarithms together like this, so we can immediately see that option B is FALSE.

STEP 5

Now, let's examine option C. According to the quotient rule of logarithms, the difference of the logarithms of two numbers is equal to the logarithm of their quotient. So, we can rewrite log9log3\log9-\log3 as log(9/3)\log (9 /3).

STEP 6

Calculate the quotient of9 and3.
9/3=39 /3 =3So, log9log3=log3\log9-\log3=\log3, which matches the given equation. Therefore, option C is TRUE.

STEP 7

Finally, let's examine option D. Using the quotient rule of logarithms, we can rewrite log6log2\log6-\log2 as log(6/2)\log (6 /2).

STEP 8

Calculate the quotient of6 and2.
6/2=36 /2 =3So, log6log2=log3\log6-\log2=\log3, not log4\log4. Therefore, option D is FALSE.
The only TRUE statement is option C loglog3=log3\log-\log3=\log3.

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