Use the properties of logarithms to find the exact value of the expression. Do not use a calculator.

log
3
⁡
24
−
log
3
⁡
8

Question

Use the properties of logarithms to find the exact value of the expression. Do not use a calculator.

log
3
⁡
24
−
log
3
⁡
8

BlackTom AI · Accepted Answer

The question asks to use the properties of logarithms to find the exact value of the expression, without a calculator.
First, recognize the subtraction of two logarithms with the same base: log_3(24) - log_3(8).
Recall the logarithm quotient rule: log_b(x) - log_b(y) = log_b(x/y). Here, b = 3, x = 24, and y = 8, so the expression becomes log_3(24/8).
Compute the fraction: 24/8 = 3. Therefore, the expression simplifies to log_3(3).
Now apply the fundamental log rule: log_b(b) = 1 for any base b > 0, b ≠ 1. Since the argument equals the base, log_3(3) = 1.
Hence, the exact value is 1.
If a student mistakenly tries to separate the logarithms or uses a different base, they might misinterpret the subtraction rule or ignore that the base must remain consistent throughout. Also, attempting to evaluate log_3(24) or log_3(8) individually without combining them would not directly yield a simple integer, illustrating why the quotient rule is essential here.

Use the properties of logarithms to find the exact value of the expression. Do not use a calculator. log 3 ⁡ 24 − log 3 ⁡ 8数值题

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