This is a logarithms question testing the product rule. Choice B (8) is correct — apply the log product rule: log₂(x) + log₂(4) = log₂(4x) = 5. Convert to exponential form: 4x = 2⁵ = 32. Solve: x = 8. Since log₂(4) = 2, you can also solve as: log₂(x) = 5 − 2 = 3 → x = 2³ = 8. Choice A (4) results from computing 2³ incorrectly — arriving at the right exponent of 3 but evaluating 2³ as 4 (possibly confusing 2² = 4 with 2³ = 8). Choice C (16) results from an off-by-one exponent error after correctly applying the product rule: correctly getting log₂(x) = 3, but then computing x = 2⁴ = 16 instead of 2³ = 8. Choice D (32) ignores the log₂(4) term entirely, solving log₂(x) = 5 → x = 2⁵ = 32. Pro tip: The log product rule states log_b(M) + log_b(N) = log_b(MN). Use it to combine the two log terms before converting to exponential form — this is almost always faster than working with them separately.

This problem uses the logarithm product property. The product property states that log_a(x) + log_a(y) = log_a(xy). Applying this property to log(2) + log(50), we get log(2) + log(50) = log(2 × 50) = log(100). Choice A incorrectly adds the arguments instead of multiplying them.

This problem uses the logarithm quotient property. The quotient property states that $\ln(a) - \ln(b) = \ln(a/b)$. Applying this property to $\ln(12) - \ln(3)$, we get $\ln(12) - \ln(3) = \ln(12/3) = \ln(4)$. Choice A incorrectly keeps the arguments separate, while choice D incorrectly reverses the fraction.

To evaluate this logarithm, we need to find what power 10 must be raised to get 1000. The logarithm property states that log_a(b) = c means $a^c$ = b. We can rewrite 1000 as $10^3$, so log₁₀(1000) = $log₁₀(10^3$). Using the power rule for logarithms, log_$a(x^n$) = n·log_a(x), we get 3·log₁₀(10) = 3·1 = 3.

To evaluate this logarithm, we need to find what power 3 must be raised to get 81. The logarithm property states that $\log_a(b) = c$ means $a^c = b$. We can rewrite 81 as $3^4$ (since $3 \cdot 3 \cdot 3 \cdot 3 = 81$). Therefore, $\log_3(81) = \log_3(3^4)$. Using the power rule, this equals $4 \cdot \log_3(3) = 4 \cdot 1 = 4$.

This problem uses the property that natural logarithm and exponential are inverse functions. The key property is that $ln(e^x$) = x for any real number x. Applying this property directly to $ln(e^5$), we get $ln(e^5$) = 5. Choice A incorrectly leaves the expression unsimplified, while choice B incorrectly multiplies by e.

Given log(x) = -2 (base 10 implied), we need to find x. Using the definition log₁₀(x) = -2 means 10^(-2) = x. Therefore, x = 10^(-2) = 1/10² = 1/100 = 0.01. Choice A gives just -2, which is the logarithm value, not x itself.

This problem uses the fundamental relationship between logarithms and exponentials. The equation log_7(49) asks "to what power must we raise 7 to get 49?" Since $7^2$ = 49, we have log_7(49) = 2. Choice A incorrectly gives the reciprocal, while choice C gives the base instead of the exponent.

This problem uses the fundamental relationship between logarithms and exponentials. The equation log_5(125) asks "to what power must we raise 5 to get 125?" Since $5^3$ = 125, we have log_5(125) = 3. Choice D incorrectly gives the argument of the logarithm rather than the exponent needed.

To solve this equation, we need to convert from logarithmic to exponential form. The equation log(x) = 2 means "10 raised to what power equals x?" Since log without a specified base typically means log₁₀, we have 10² = x. Therefore, x = 100. Choice B would give 10¹ = 10, which doesn't satisfy the original equation.