Solve for $$ x $$ : $$ \log _{5} x+\log _{5}(x+2)=7 $$ $$ x=\square $$ Question Help: Video Submit Question

Question

UpStudy AI · Accepted Answer

We start with the equation

$$
\log_{5}x+\log_{5}(x+2)=7.
$$

**Step 1. Combine the logarithms**

Using the logarithm property

$$
\log_{a}u+\log_{a}v=\log_{a}(uv),
$$

we can combine the two logarithms:

$$
\log_{5}(x(x+2))=7.
$$

**Step 2. Convert to exponential form**

Recalling that

$$
\log_{a}b=c\quad\Longleftrightarrow\quad b=a^c,
$$

we rewrite the equation as

$$
x(x+2)=5^7.
$$

We know that

$$
5^7=78125,
$$

so our equation becomes

$$
x(x+2)=78125.
$$

**Step 3. Write and solve the quadratic equation**

Expanding the left-hand side gives

$$
x^2+2x=78125.
$$

Subtract $$78125$$ from both sides to set the equation to zero:

$$
x^2+2x-78125=0.
$$

We solve this quadratic equation using the quadratic formula

$$
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},
$$

where $$a=1$$, $$b=2$$, and $$c=-78125$$.

Substitute the values into the formula:

$$
x=\frac{-2\pm\sqrt{2^2-4(1)(-78125)}}{2(1)}=\frac{-2\pm\sqrt{4+312500}}{2}=\frac{-2\pm\sqrt{312504}}{2}.
$$

**Step 4. Choose the valid solution**

Since logarithms are defined only for positive arguments, we require $$x>0$$. Therefore, we discard the negative solution and keep:

$$
x=\frac{-2+\sqrt{312504}}{2}.
$$

This is the exact solution for $$x$$.

Thus, the final answer is

$$
x=\frac{-2+\sqrt{312504}}{2}.
$$
$$ x = \frac{-2 + \sqrt{312504}}{2} $$

Solve for \( x \) : \( \log _{5} x+\log _{5}(x+2)=7 \) \( x=\square \) Question Help: Video Submit Question