To solve for [tex]\( x \)[/tex] in the equation
[tex]\[ 3 \log_2(x + 18) = 12, \][/tex]
we'll go through the following steps:
Step 1: Isolate the logarithm
First, divide both sides of the equation by 3 to isolate the logarithmic term. This simplifies to:
[tex]\[ \log_2(x + 18) = \frac{12}{3} \][/tex]
[tex]\[ \log_2(x + 18) = 4 \][/tex]
Step 2: Convert from logarithmic to exponential form
To remove the logarithm, rewrite the equation in its exponential form. The equation [tex]\(\log_b(A) = C\)[/tex] can be rewritten as [tex]\(A = b^C\)[/tex]. In our case, [tex]\(b\)[/tex] is 2, [tex]\(A\)[/tex] is [tex]\(x + 18\)[/tex], and [tex]\(C\)[/tex] is 4:
[tex]\[ x + 18 = 2^4 \][/tex]
[tex]\[ x + 18 = 16 \][/tex]
Step 3: Solve for [tex]\( x \)[/tex]
Now, solve for [tex]\( x \)[/tex] by isolating it on one side of the equation. Subtract 18 from both sides:
[tex]\[ x = 16 - 18 \][/tex]
[tex]\[ x = -2 \][/tex]
Therefore, the solution to the equation [tex]\( 3 \log_2(x + 18) = 12 \)[/tex] is:
[tex]\[ x = -2 \][/tex]
