Express the solution of 2^(x+5) = 2 + 2^x in logarithmic form.

Express the solution of 2^(x+5) = 2 + 2^x in logarithmic form.

• A. log_2(31/2)
• B. log_2(2/31)
• C. log_2(2^x)
• D. log_2(2 + 2^x)

Answer: (B)

To explore why the correct answer can be rooted in logarithmic properties, let's first manipulate the equation given in the question: Starting with \( 2^{(x+5)} = 2 + 2^{x} \), we can simplify this to better understand its logarithmic representation. First, we rewrite the left side: - \( 2^{(x+5)} \) can be expressed as \( 2^x \cdot 2^5 \) or \( 32 \cdot 2^x \). The equation then transforms to: - \( 32 \cdot 2^x = 2 + 2^x \). Next, we can rewrite the equation by moving all terms involving \( 2^x \) to one side: - \( 32 \cdot 2^x - 2^x = 2 \), which simplifies to - \( 31 \cdot 2^x = 2 \). From here, we divide both sides by 31: - \( 2^x = \frac{2}{31} \). At this point, we can express \( x \) in logarithmic form. To isolate \( x \

To explore why the correct answer can be rooted in logarithmic properties, let's first manipulate the equation given in the question:

Starting with ( 2^{(x+5)} = 2 + 2^{x} ), we can simplify this to better understand its logarithmic representation.

First, we rewrite the left side:

• ( 2^{(x+5)} ) can be expressed as ( 2^x \cdot 2^5 ) or ( 32 \cdot 2^x ).

The equation then transforms to:

• ( 32 \cdot 2^x = 2 + 2^x ).

Next, we can rewrite the equation by moving all terms involving ( 2^x ) to one side:

• ( 32 \cdot 2^x - 2^x = 2 ),

which simplifies to

• ( 31 \cdot 2^x = 2 ).

From here, we divide both sides by 31:

• ( 2^x = \frac{2}{31} ).

At this point, we can express ( x ) in logarithmic form. To isolate ( x \