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.

To convert from logarithmic to exponential form, we use the fundamental property that log_a(b) = c is equivalent to $a^c$ = b. In the equation log₃(27) = y, the base is 3, the argument is 27, and the result is y. Converting to exponential form gives us $3^y$ = 27. Choice B incorrectly reverses the base and argument positions.