Let's solve the given equation step by step.
We start with the equation:
[tex]\[ 2^{x+1} + 2^x - 2^{x-1} = 28 \][/tex]
We will use properties of exponents to simplify this equation.
First, let's express each term using properties of exponents:
- [tex]\( 2^{x+1} \)[/tex] can be rewritten as [tex]\( 2 \cdot 2^x \)[/tex] because [tex]\( 2^{x+1} = 2^x \cdot 2^1 = 2 \cdot 2^x \)[/tex].
- [tex]\( 2^{x-1} \)[/tex] can be rewritten as [tex]\( \frac{2^x}{2} \)[/tex] because [tex]\( 2^{x-1} = 2^x \cdot 2^{-1} = \frac{2^x}{2} \)[/tex].
Substituting these into the original equation:
[tex]\[ 2 \cdot 2^x + 2^x - \frac{2^x}{2} = 28 \][/tex]
Now, let's combine like terms:
[tex]\[ 2 \cdot 2^x + 2^x = 3 \cdot 2^x \][/tex]
[tex]\[ 3 \cdot 2^x - \frac{2^x}{2} = 28 \][/tex]
The term [tex]\( \frac{2^x}{2} \)[/tex] is equal to [tex]\( 0.5 \cdot 2^x \)[/tex], so we can rewrite the equation as:
[tex]\[ 3 \cdot 2^x - 0.5 \cdot 2^x = 28 \][/tex]
Combine the coefficients of [tex]\( 2^x \)[/tex]:
[tex]\[ (3 - 0.5) \cdot 2^x = 28 \][/tex]
[tex]\[ 2.5 \cdot 2^x = 28 \][/tex]
To isolate [tex]\( 2^x \)[/tex], divide both sides of the equation by 2.5:
[tex]\[ 2^x = \frac{28}{2.5} \][/tex]
[tex]\[ 2^x = 11.2 \][/tex]
Now we need to solve for [tex]\( x \)[/tex]. To do this, we need to take the logarithm of both sides. We typically use logarithm base 2 for problems involving powers of 2, but logarithm base 10 or the natural logarithm (base [tex]\( e \)[/tex]) can also work with appropriate conversions. Here, we are using the properties of natural logarithms (base [tex]\( e \)[/tex]):
[tex]\[ x = \log_2(11.2) \][/tex]
This expression can be converted to use the natural logarithm:
[tex]\[ x = \frac{\ln(11.2)}{\ln(2)} \][/tex]
Given the preceding steps and conversion:
[tex]\[ x = \frac{\ln(56/5)}{\ln(2)} \][/tex]
The solution can be expressed as:
[tex]\[ x = \frac{-\ln(5) + \ln(56)}{\ln(2)} \][/tex]
Thus, the value of [tex]\( x \)[/tex] is:
[tex]\[ x = \frac{\ln(56) - \ln(5)}{\ln(2)} \][/tex]
This is the precise solution to the given equation [tex]\( 2^{x+1} + 2^x - 2^{x-1} = 28 \)[/tex].
