Logarithmic Expressions Review

log3(1/81) = log3(3^-4) = -4 and log(10^-4) = -4. Thus, these two expressions equal -4.

Logarithms are undefined for non-positive numbers. log2 0 is undefined because you cannot take the log of zero, and log2 (−16) is undefined because the argument is negative. Thus, the correct answers are log2 0 and log2 (−16).

If log15 b = 0, it means 15 raised to the power of 0 equals b. Since any number to the power of 0 is 1, b must be 1. Therefore, the correct answer is 1.

The logarithm log5 5 equals 1 because the logarithm of a number to its own base is always 1. The other options do not equal 1: log5 1 equals 0, log5 0 is undefined, and log5 (1/5) is negative.

Using the properties of logarithms, specifically the product property, we can expand log(9n) as log(9) + log(n). Therefore, the correct choice is log 9 + log n.

To expand log 5x^3y, use the properties of logarithms: log(ab) = log a + log b and log(a^b) = b log a. Thus, log 5x^3y = log 5 + log(x^3) + log y = log 5 + 3 log x + log y. The correct choice is log 5 + 3 log x + log y.

Using the properties of logarithms, specifically the quotient rule, we can expand log(c/45) as log(c) - log(45). Thus, the correct choice is log(c) - log(45).

To expand \log \left( \frac{10x^3}{y^4} \right), use the properties: \log \left( \frac{a}{b} \right) = \log a - \log b \text{ and } \log a^n = n \log a. This gives \log 10 + 3 \log x - 4 \log y, which matches the correct choice.

Using the properties of logarithms, we expand \log \left(\frac{4x^3}{11y^2}\right) = \log(4) + \log(x^3) - \log(11) - \log(y^2). Then, applying the power rule gives us \log(4) + 3\log(x) - \log(11) - 2\log(y), which is the correct choice.

Using the power rule of logarithms, we expand log(3x^4y^2)^4 as 4 * log(3x^4y^2). Then, applying the product and power rules, we get 4log(3) + 4*4log(x) + 4*2log(y), simplifying to 4log(3) + 16log(x) + 8log(y). Thus, the correct choice is 4log(3) + 16log(x) + 8log(y).