Question

Lg 5+(1/2)lg(x+5)=lg 3+lg(x+1)

Answer 1

                                              Решение:

Смотри вложение:...........

Answer 2

To solve the equation $5 + \frac{1}{2}\lg(x+5) = \lg 3 + \lg(x+1)$, you can start by simplifying and rearranging the terms. The goal is to isolate the variable, $x$.

First, let's work on the equation step by step:

1. Subtract $\lg 3$ from both sides of the equation to isolate the logarithmic terms:

   $5 + \frac{1}{2}\lg(x+5) - \lg 3 = \lg(x+1)$

2. Combine the constants on the left side:

   $5 - \lg 3 + \frac{1}{2}\lg(x+5) = \lg(x+1)$

3. Next, subtract $\frac{1}{2}\lg(x+5)$ from both sides to isolate $\lg(x+1)$ on the right side:

   $5 - \lg 3 = \lg(x+1) - \frac{1}{2}\lg(x+5)$

4. Now, we can combine the logarithmic terms on the right side by using the properties of logarithms. Recall that subtracting logarithms is equivalent to dividing the numbers they are applied to:

   $5 - \lg 3 = \lg\left(\frac{x+1}{\sqrt{x+5}}\right)$

5. To eliminate the logarithm on the right side, you can rewrite the equation in exponential form:

   $10^{5 - \lg 3} = \frac{x+1}{\sqrt{x+5}}$

6. Solve for $x$. First, calculate $10^{5 - \lg 3}$:

   $10^{5 - \lg 3} \approx 291.5$

7. Now, multiply both sides by $\sqrt{x+5}$ to get rid of the square root:

   $291.5\sqrt{x+5} = x+1$

8. To isolate $x$, subtract $x$ from both sides and subtract 1 from both sides:

   $291.5\sqrt{x+5} - x = 1$

9. You may need to use numerical methods or an online equation solver to find the exact value of $x$ because this is a nonlinear equation.

So, the solution for $x$ is an approximation, and you can use a numerical solver to find the exact value.