The equation you provided is:

$\log_{10}(x+2) + \log_{10}(x) - 1 = \log_{10}\left(\frac{3}{2}\right)$

We can solve this step-by-step by following logarithmic properties.

Step 1: Apply the logarithmic identity

Recall that the sum of logarithms can be combined as the logarithm of the product: $\log_{10}(a) + \log_{10}(b) = \log_{10}(a \cdot b)$ So, we can combine the first two terms: $\log_{10}((x+2) \cdot x) - 1 = \log_{10}\left(\frac{3}{2}\right)$ This simplifies to: $\log_{10}(x(x+2)) - 1 = \log_{10}\left(\frac{3}{2}\right)$

Step 2: Eliminate the constant term

Now, isolate the logarithmic term on one side. We know that subtracting 1 is equivalent to subtracting $\log_{10}(10)$ (since $\log_{10}(10) = 1$): $\log_{10}(x(x+2)) = \log_{10}\left(\frac{3}{2}\right) + \log_{10}(10)$ Simplify the right-hand side using the logarithmic property again: $\log_{10}(x(x+2)) = \log_{10}\left(10 \cdot \frac{3}{2}\right)$ This becomes: $\log_{10}(x(x+2)) = \log_{10}(15)$

Step 3: Set the arguments equal

Since both sides are logarithms with the same base, we can equate the arguments: $x(x+2) = 15$

Step 4: Solve the quadratic equation

Expand and simplify: $x^2 + 2x = 15$ $x^2 + 2x - 15 = 0$

Now, solve this quadratic equation using factoring, the quadratic formula, or completing the square. Factoring gives: $(x + 5)(x - 3) = 0$ So, the solutions are: $x = -5 \quad \text{or} \quad x = 3$

Step 5: Check for valid solutions

We must check that both solutions are valid by substituting them back into the original logarithmic equation. Logarithms require the argument to be positive, so we cannot have negative values inside a logarithm.

• For $x = -5$, both $\log_{10}(x+2)$ and $\log_{10}(x)$ would be undefined because logarithms of negative numbers are not real.
• For $x = 3$, both $\log_{10}(x+2)$ and $\log_{10}(x)$ are defined.

Thus, the only valid solution is: $x = 3$

Let me know if you need further clarification or any additional details!

Questions to explore further:

1. How do logarithmic properties help in solving equations?
2. How can you use the quadratic formula to solve equations?
3. What is the domain of a logarithmic function?
4. How do you solve equations involving logarithms with different bases?
5. What happens when a logarithmic equation has no solution?

Tip: Always check the domain when solving logarithmic equations, as negative values inside a logarithm are undefined in real numbers.