Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = \frac{3}{5} \)
Solution
To solve the equation \( \log _{5}(5x + 2) = 1 \), follow these steps:
1. **Understand the logarithmic equation**: The equation \( \log_b(a) = c \) can be rewritten in its exponential form as \( b^c = a \).
2. **Apply the exponential form**:
\[
\log _{5}(5x + 2) = 1 \quad \Rightarrow \quad 5^1 = 5x + 2
\]
Simplifying the left side:
\[
5 = 5x + 2
\]
3. **Solve for \( x \)**:
\[
5x + 2 = 5
\]
Subtract 2 from both sides:
\[
5x = 3
\]
Divide both sides by 5:
\[
x = \frac{3}{5}
\]
**Final Answer:**
\( x = \dfrac{3}{5} \)
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Simplify this solution Bonus Knowledge
To solve the equation \( \log_{5}(5x + 2) = 1 \), we start by using the definition of logarithms. The equation states that \( 5^{1} = 5x + 2 \). This simplifies to: \[ 5 = 5x + 2 \] Next, we subtract 2 from both sides: \[ 5 - 2 = 5x \] This gives us: \[ 3 = 5x \] Now, we divide both sides by 5: \[ x = \frac{3}{5} \] Therefore, the solution to the equation \( \log_{5}(5x + 2) = 1 \) is \( x = \frac{3}{5} \).
