Sure, let's solve the equation [tex]\(\log_2 x - \log_2(x + 3) = -21\)[/tex] step-by-step.
1. Recall the properties of logarithms:
Using the property of logarithms that states [tex]\(\log_a b - \log_a c = \log_a \left(\frac{b}{c}\right)\)[/tex], we can rewrite the left-hand side of the equation.
[tex]\[
\log_2 x - \log_2 (x + 3) = \log_2 \left(\frac{x}{x + 3}\right)
\][/tex]
Thus, the equation becomes:
[tex]\[
\log_2 \left(\frac{x}{x + 3}\right) = -21
\][/tex]
2. Exponentiate both sides to remove the logarithm:
To eliminate the logarithm, we raise 2 to the power of both sides of the equation, using the fact [tex]\(a^{\log_a b} = b\)[/tex]. Hence:
[tex]\[
2^{\log_2 \left(\frac{x}{x + 3}\right)} = 2^{-21}
\][/tex]
This simplifies to:
[tex]\[
\frac{x}{x + 3} = 2^{-21}
\][/tex]
3. Solve the resulting equation:
To solve for [tex]\(x\)[/tex], multiply both sides by [tex]\(x + 3\)[/tex] to clear the fraction:
[tex]\[
x = 2^{-21} (x + 3)
\][/tex]
Distribute [tex]\(2^{-21}\)[/tex] on the right-hand side:
[tex]\[
x = \frac{1}{2^{21}} x + \frac{3}{2^{21}}
\][/tex]
4. Isolate [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], first move the [tex]\(\frac{1}{2^{21}} x\)[/tex] term to the left-hand side:
[tex]\[
x - \frac{1}{2^{21}} x = \frac{3}{2^{21}}
\][/tex]
Factor [tex]\(x\)[/tex] out of the left-hand side:
[tex]\[
x \left(1 - \frac{1}{2^{21}}\right) = \frac{3}{2^{21}}
\][/tex]
5. Simplify the expression:
Recognize that [tex]\(1 - \frac{1}{2^{21}} = \frac{2^{21} - 1}{2^{21}}\)[/tex], so the equation becomes:
[tex]\[
x \cdot \frac{2^{21} - 1}{2^{21}} = \frac{3}{2^{21}}
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], multiply both sides by [tex]\(\frac{2^{21}}{2^{21} - 1}\)[/tex]:
[tex]\[
x = \frac{3}{2^{21} - 1}
\][/tex]
Simplify to get:
[tex]\[
x = \frac{3}{2^{21} - 1} = \frac{3}{2097151}
\][/tex]
So, the solution to the equation [tex]\(\log_2 x - \log_2(x + 3) = -21\)[/tex] is:
[tex]\[
x = \frac{3}{2097151}
\][/tex]
