To solve the equation [tex]\( \log_2(x + 9) = 3 + \log_2(x + 2) \)[/tex], we can follow these steps:
1. Step 1: Simplify the equation
Start with the equation:
[tex]\[
\log_2(x + 9) = 3 + \log_2(x + 2)
\][/tex]
Subtract [tex]\(\log_2(x + 2)\)[/tex] from both sides to isolate the logarithms:
[tex]\[
\log_2(x + 9) - \log_2(x + 2) = 3
\][/tex]
2. Step 2: Apply the properties of logarithms
Use the property of logarithms that [tex]\(\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)\)[/tex]:
[tex]\[
\log_2\left(\frac{x + 9}{x + 2}\right) = 3
\][/tex]
3. Step 3: Convert the logarithmic equation to an exponential equation
Recall that if [tex]\(\log_b(y) = c\)[/tex], then [tex]\(y = b^c\)[/tex]. In this case, set the argument inside the logarithm equal to [tex]\(2^3\)[/tex] (since the base is 2):
[tex]\[
\frac{x + 9}{x + 2} = 2^3
\][/tex]
Simplify the exponent:
[tex]\[
\frac{x + 9}{x + 2} = 8
\][/tex]
4. Step 4: Solve for [tex]\(x\)[/tex]
Set up the equation:
[tex]\[
x + 9 = 8(x + 2)
\][/tex]
Distribute the right side:
[tex]\[
x + 9 = 8x + 16
\][/tex]
Move all terms involving [tex]\(x\)[/tex] to one side of the equation:
[tex]\[
x + 9 - 8x = 16
\][/tex]
Simplify:
[tex]\[
-7x + 9 = 16
\][/tex]
Subtract 9 from both sides:
[tex]\[
-7x = 7
\][/tex]
Divide both sides by -7:
[tex]\[
x = -1
\][/tex]
5. Step 5: Verify the solution
Substitute [tex]\(x = -1\)[/tex] back into the original equation to ensure it satisfies it:
[tex]\[
\log_2((-1) + 9) = 3 + \log_2((-1) + 2)
\][/tex]
Simplify inside the logarithms:
[tex]\[
\log_2(8) = 3 + \log_2(1)
\][/tex]
We know that:
[tex]\[
\log_2(8) = 3 \quad \text{and} \quad \log_2(1) = 0
\][/tex]
Therefore:
[tex]\[
3 = 3 + 0 \Rightarrow 3 = 3
\][/tex]
The equality holds true, confirming that our solution is correct.
Therefore, the solution to the equation is:
[tex]\[
x = -1
\][/tex]
