To solve the equation $\log(6x + 4) = 5 + \log(2x - 5)$ for $x$, we can proceed as follows:

1. Rewrite the equation: Move all logarithmic terms to one side of the equation: $\log(6x + 4) - \log(2x - 5) = 5$

2. Apply the properties of logarithms: Use the quotient rule for logarithms, $\log(a) - \log(b) = \log\left(\frac{a}{b}\right)$, to combine the logs: $\log\left(\frac{6x + 4}{2x - 5}\right) = 5$

3. Rewrite in exponential form: To remove the logarithm, rewrite the equation in exponential form. Since the base of a common logarithm is 10, we get: $\frac{6x + 4}{2x - 5} = 10^5$ Simplify $10^5$ as $100000$: $\frac{6x + 4}{2x - 5} = 100000$

4. Cross-multiply to clear the fraction: $6x + 4 = 100000 \cdot (2x - 5)$ Expanding on the right: $6x + 4 = 200000x - 500000$

5. Isolate $x$: Move all terms involving $x$ to one side of the equation and constant terms to the other: $4 + 500000 = 200000x - 6x$ Simplify: $500004 = 199994x$

6. Solve for $x$: $x = \frac{500004}{199994} = 2.50005$

7. Check for extraneous solutions: Substitute $x = 2.50005$ back into the original expressions inside the logarithms to ensure they are positive:

   • $6x + 4 = 6(2.50005) + 4 = 15.0003 + 4 = 19.0003$, which is positive.
   • $2x - 5 = 2(2.50005) - 5 = 5.0001 - 5 = 0.0001$, which is also positive.

Since both expressions inside the logarithms are positive, $x = 2.50005$ is valid.

Final Answer:

$x = 2.50005$

Would you like further explanation on any step, or have questions about this problem?

Here are five related questions for further exploration:

1. How can you apply logarithmic properties in equations involving different bases?
2. What are extraneous solutions, and why are they important in logarithmic equations?
3. How would this problem differ if a different base were used?
4. Why do we need to ensure that terms inside logarithms remain positive?
5. What is the process of converting between logarithmic and exponential forms?

Tip: Always check your solution back in the original equation to avoid extraneous solutions, especially with logarithmic functions where domains are restricted to positive values.