To simplify the expression $\log_x{5} - 3\log_x{2} + \log_x{4}$, we can apply the logarithmic properties. Here’s the step-by-step solution:

1. Apply the logarithmic property for subtraction:

$\log_x{a} - \log_x{b} = \log_x{\left(\frac{a}{b}\right)}$

First, apply this property to the first two terms: $\log_x{5} - 3\log_x{2} = \log_x{\left(\frac{5}{2^3}\right)} = \log_x{\left(\frac{5}{8}\right)}$

2. Use the power rule of logarithms:

$a\log_x{b} = \log_x{b^a}$ This has already been applied to the term $3\log_x{2}$ to get $2^3 = 8$.

3. Add the third term $\log_x{4}$ using the logarithmic property of addition:

$\log_x{\left(\frac{5}{8}\right)} + \log_x{4} = \log_x{\left(\frac{5}{8} \cdot 4\right)}$

4. Simplify the expression:

$\frac{5}{8} \cdot 4 = \frac{5}{2}$

Thus, the simplified expression is: $\log_x{\left(\frac{5}{2}\right)}$

Would you like more details on any part of this explanation?

5 Related Questions:

1. What are the basic properties of logarithms?
2. How does the power rule of logarithms work?
3. Can logarithms with different bases be simplified?
4. How do you solve logarithmic equations?
5. What is the change of base formula for logarithms?

Tip: Always remember to use the logarithmic properties (product, quotient, and power rules) when simplifying complex logarithmic expressions.