To solve for $x$ in the equation

$\log(x + 10) = \log(x) + \log(5),$

we can use the properties of logarithms to simplify and isolate $x$.

Step 1: Apply the Product Rule

The right side of the equation can be simplified using the product rule for logarithms, which states that $\log(a) + \log(b) = \log(a \cdot b)$.

$\log(x) + \log(5) = \log(5x)$

So the equation becomes:

$\log(x + 10) = \log(5x)$

Step 2: Remove the Logarithms

Since the logs on both sides of the equation are equal, we can set the arguments equal to each other:

$x + 10 = 5x$

Step 3: Solve for $x$

Now, isolate $x$ by moving all terms involving $x$ to one side of the equation:

$10 = 5x - x$ $10 = 4x$ $x = \frac{10}{4} = \frac{5}{2}$

Conclusion

The solution is:

$x = \frac{5}{2}$

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If you have any questions or would like a breakdown of any step, feel free to ask!

Related Questions

1. How do you solve logarithmic equations with multiple logarithmic terms?
2. Can you show how to solve $\log(x^2) = 2 \log(x)$ without using the product rule?
3. How does the product rule for logarithms apply to negative values of $x$?
4. What happens if $x = -5$ in this equation?
5. Can the same approach be used for equations like $\log(x+2) = \log(x) + \log(3)$?

Tip

When dealing with logarithmic equations, look out for properties like the product and quotient rules to simplify expressions before isolating variables.