Let’s solve each of these logarithmic equations step by step.

1. $\log_x 16 = 2$

This equation translates to: $x^2 = 16$ Taking the square root of both sides, we get: $x = \pm 4$ So, $x = 4$ (assuming we're working with positive bases in typical logarithmic contexts).

2. $\log_{27} 3 = y$

This equation implies: $27^y = 3$ Since $27 = 3^3$, we can rewrite this as: $(3^3)^y = 3 \Rightarrow 3^{3y} = 3^1$ Equating the exponents: $3y = 1 \Rightarrow y = \frac{1}{3}$

3. $\log_3 (2x - 1) = 2$

Rewrite the equation as: $3^2 = 2x - 1$ This simplifies to: $9 = 2x - 1$ Adding 1 to both sides: $10 = 2x \Rightarrow x = 5$

4. $\log_2 x = -3$

Rewrite the equation as: $2^{-3} = x$ This gives: $x = \frac{1}{8}$

5. $\log_3 (5x + 1) = 4$

Rewrite this as: $3^4 = 5x + 1$ Calculating $3^4$: $81 = 5x + 1$ Subtracting 1 from both sides: $80 = 5x \Rightarrow x = 16$

Summary of Solutions:

1. $x = 4$
2. $y = \frac{1}{3}$
3. $x = 5$
4. $x = \frac{1}{8}$
5. $x = 16$

Would you like further details on any of these solutions, or do you have any questions?

Related Questions

1. How would you solve a logarithmic equation with a variable in both the base and the argument?
2. What are the properties of logarithms that simplify solving logarithmic equations?
3. How can logarithmic equations be applied in real-world contexts, such as in exponential growth?
4. What methods can you use to solve a logarithmic equation with different bases on each side?
5. How do you handle logarithmic equations that involve multiple logarithms on one side?

Tip

Always check the domain of logarithmic functions to ensure that all arguments and bases are valid (positive and non-zero).