Skill: Solve logarithmic equations

To solve a logarithmic equation:

1. State the domain: Every logarithm’s argument must be positive.
2. Use logarithm properties to combine or simplify logarithms.
3. Rewrite in exponential form or solve the resulting algebraic equation.
4. Check all answers in the original equation, because algebra can produce values outside the logarithm’s domain.

Example

Solve:

$$\log_2(x)+\log_2(x-2)=3$$

Step 1: Find the domain.

Both logarithm arguments must be positive:

$$x>0 \qquad \text{and} \qquad x-2>0$$

Together, these give:

$$x>2$$

Step 2: Combine the logarithms.

Using the product property,

$$\log_b M+\log_b N=\log_b(MN)$$

we get:

$$\log_2\bigl(x(x-2)\bigr)=3$$

Step 3: Rewrite in exponential form.

Since $\log_b A=c$ means $A=b^c$,

$$x(x-2)=2^3$$

So:

$$x^2-2x=8$$ $$x^2-2x-8=0$$

Step 4: Solve the quadratic.

Factor:

$$(x-4)(x+2)=0$$

Therefore,

$$x=4 \qquad \text{or} \qquad x=-2$$

Step 5: Check the domain.

The domain requires $x>2$. Thus, $x=-2$ is not allowed.

Check $x=4$ in the original equation:

$$\log_2(4)+\log_2(2)=2+1=3$$

Therefore, the solution is:

$$\boxed{x=4}$$