To solve the equation [tex]\( 2^x + 2^{x-1} - 48 = 0 \)[/tex], let's follow these logical steps.
1. Rewrite the Equation:
Notice that [tex]\( 2^{x-1} \)[/tex] can be written as [tex]\( \frac{2^x}{2} \)[/tex]. Therefore, the given equation can be rewritten as:
[tex]\[
2^x + \frac{2^x}{2} - 48 = 0
\][/tex]
2. Combine Like Terms:
Combine the terms involving [tex]\( 2^x \)[/tex]. We express [tex]\( 2^x + \frac{2^x}{2} \)[/tex] with a common factor:
[tex]\[
2^x + \frac{2^x}{2} = 2^x \left(1 + \frac{1}{2}\right) = 2^x \left(\frac{3}{2}\right)
\][/tex]
Hence, the equation becomes:
[tex]\[
\frac{3}{2} \cdot 2^x - 48 = 0
\][/tex]
3. Isolate the Exponential Expression:
To isolate [tex]\( 2^x \)[/tex], multiply both sides of the equation by [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[
2^x = 48 \cdot \frac{2}{3}
\][/tex]
Simplifying the right-hand side:
[tex]\[
2^x = 32
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
We know that [tex]\( 32 \)[/tex] can be expressed as a power of [tex]\( 2 \)[/tex]. Specifically:
[tex]\[
32 = 2^5
\][/tex]
Thus, we have:
[tex]\[
2^x = 2^5
\][/tex]
5. Equating the Exponents:
Since the bases are the same, we can equate the exponents:
[tex]\[
x = 5
\][/tex]
Therefore, the solution to the equation [tex]\( 2^x + 2^{x-1} - 48 = 0 \)[/tex] is:
[tex]\[
x = 5
\][/tex]
