Question
If \( \log _{2}(2 x+10)=5 \), then what is the value of \( x \) ? \( \begin{array}{ll}\text { a.) }-5 \\ \text { b.) } 10 \\ \text { c.) } 11 \\ \text { d.) }-3\end{array} \)
Ask by Cervantes Blake. in the United States
Mar 01,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = 11 \), choice c.
Solution
We are given the equation log₂(2x + 10) = 5. Recall that the definition of logarithms states logₐ(b) = c means that aᶜ = b. Therefore, we can rewrite the equation as:
2⁵ = 2x + 10
Calculating 2⁵ gives 32:
32 = 2x + 10
Subtract 10 from both sides to isolate the term with x:
32 - 10 = 2x
22 = 2x
Now, divide both sides by 2:
x = 11
Thus, the value of x is 11, which corresponds to option c.
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Simplify this solution Bonus Knowledge
To solve the equation \( \log_{2}(2x + 10) = 5 \), we first rewrite it in exponential form. This gives us: \[ 2x + 10 = 2^5 \] Calculating \( 2^5 \), we find that \( 2^5 = 32 \). Therefore, we have: \[ 2x + 10 = 32 \] Next, we can isolate \( 2x \) by subtracting 10 from both sides: \[ 2x = 32 - 10 \] \[ 2x = 22 \] Now, divide both sides by 2 to find \( x \): \[ x = \frac{22}{2} = 11 \] Thus, the value of \( x \) is \( 11 \). The correct answer is \( c.) 11 \).
