Write each expression as a single logarithm.

log2(
1

x
)+log2(
1

x2
)

*Write log5x as log5(x) **Write x4 as x^4

Question

Write each expression as a single logarithm.

log2(
1

x
)+log2(
1

x2
)

*Write log5x as log5(x)  **Write x4 as x^4

BlackTom AI · Accepted Answer

To begin, I’ll restate what the task asks: combine the two logarithmic expressions into a single logarithm.
Option assessment is tricky here because there aren’t multiple-choice distractors; instead, the provided answer options show two mathematically equivalent single-log forms.
First, consider the given expression: log2(1/x) + log2(1/x^2).
Using the product rule for logs, log_a(b) + log_a(c) = log_a(bc), we combine: log2(1/x) + log2(1/x^2) = log2((1/x) * (1/x^2)).
Compute the product inside the log: (1/x) * (1/x^2) = 1/x^3 = x^-3.
Thus, log2((1/x) * (1/x^2)) = log2(1/x^3) = log2(x^-3).
Now, addressing the two presented forms:
- log2(1/x^3): This is the direct result of combining the two logs under the product rule, with the inside written as a reciprocal power.
- log2(x^-3): This is the same quantity rewritten using the power rule for logarithms, since log_a(b^k) = k * log_a(b) implies log_a(1/x^3) = log_a(x^-3) because 1/x^3 = x^-3.
Both expressions are algebraically identical; they differ only in the chosen representation of the inside of the log.
If a student prefers positive exponents, log2(1/x^3) is often chosen, whereas log2(x^-3) might appeal to those comfortable with negative exponents.
Common pitfalls to avoid: assuming the sum of logs equals log of a sum, which would be incorrect; here we correctly apply the product rule, not a sum rule.
In summary, the two forms log2(1/x^3) and log2(x^-3) are equivalent single-log expressions for the given sum, and either can be used depending on the preferred inside-the-log representation.

MATH136-1IP-IO-202430-I-81X M08: Final Exam Review Quiz - Timed

Write each expression as a single logarithm. log2( 1 x )+log2( 1 x2 ) *Write log5x as log5(x) **Write x4 as x^4

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