Solve the following logarithmic expressions:

Okay, let's solve the logarithmic expressions step by step. a) log₂(1/3) + log₂(9) + log₅₃(3√2) + log₃(1/2) First, let's use the logarithm property that states log_a(b) + log_a(c) = log_a(b*c). So, we combine the first two terms: log₂(1/3) + log₂(9) = log₂((1/3) * 9) = log₂(3) Now, let's rewrite the third term. It seems like there is a typo here. Assuming it is log₅(3√2), we will keep it as is since we can't simplify it further without additional information or changing the base. Let's simplify the last term. We can write 1/2 as 2^-1. Therefore: log₃(1/2) = log₃(2^-1) = -log₃(2) Combining all these terms, we get: log₂(3) + log₅(3√2) - log₃(2) Since we cannot simplify this expression to a single number without using a calculator or more advanced techniques (like change of base formula combined with approximations), let's consider another possible initial expression. If we assume it was log_3√2(1/2), then: log_3√2(1/2) = log_3√2(2^-1) = -log_3√2(2) And if the third initial term was log₂(3√2), we have: log₂(3√2) = log₂(3 * 2^1/2) = log₂(3) + log₂(2^1/2) = log₂(3) + 1/2 So initial expression becomes: log₂(3) + log₂(3√2) - log₃(2) = log₂(3) + log₂(3) + 1/2 - log₃(2) = 2log₂(3) + 1/2 - log₃(2) Without more context, it's hard to determine what the original intended expression was. b) 5 * log₂(9) * log₈(64) Let's start by simplifying log₈(64). Since 8² = 64, we have: log₈(64) = 2 Now, we substitute this value back into the original expression: 5 * log₂(9) * 2 = 10 * log₂(9) Since 9 = 3², we can write: 10 * log₂(3²) = 10 * 2 * log₂(3) = 20 * log₂(3) So, the simplified expression is 20log₂(3) c) 5 * 2 * log₂(64) / log₂(2) First, simplify log₂(64). Since 2⁶ = 64: log₂(64) = 6 Also, log₂(2) = 1 Now substitute these values back into the expression: 5 * 2 * 6 / 1 = 10 * 6 = 60 d) 2^4log₂(3) + log₃(64) + log₃(1) Using the power rule of logarithms, a^log_a(b) = b: 2^4log₂(3) can be written as (2^log₂(3))⁴ = 3⁴ = 81 Also, log₃(1) = 0 (because any number to the power of 0 is 1) So, the expression becomes: 81 + log₃(64) + 0 = 81 + log₃(64) 64 isn't a power of 3, so we leave this as is. e) 0.6 * 81 + log₅(40.5) + 5.3 First, let's compute 0.6 * 81: 0.6 * 81 = 48.6 So the expression becomes: 48.6 + log₅(40.5) + 5.3 = 53.9 + log₅(40.5) Since 40.5 isn't a simple power of 5, we leave this as is. In summary: * a) log₂(3) + log₅(3√2) - log₃(2) *OR* 2log₂(3) + 1/2 - log₃(2) * b) 20log₂(3) * c) 60 * d) 81 + log₃(64) * e) 53.9 + log₅(40.5)