Pregunta
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\( x = 4 \)
Solución
To solve the logarithmic equation:
\[
\log_{3}(x + 5) = 2
\]
**Step 1: Understand the Logarithmic Equation**
The equation \(\log_{3}(x + 5) = 2\) asks the question: "To what power must 3 be raised to obtain \(x + 5\)?"
Mathematically, this can be rewritten using the definition of a logarithm:
\[
3^{2} = x + 5
\]
**Step 2: Solve for \(x\)**
Calculate \(3^{2}\):
\[
3^{2} = 9
\]
So,
\[
9 = x + 5
\]
Subtract 5 from both sides to isolate \(x\):
\[
x = 9 - 5
\]
\[
x = 4
\]
**Step 3: Verify the Solution**
It's important to ensure that the solution makes the original logarithmic expression valid. The argument of a logarithm must be positive:
\[
x + 5 > 0
\]
\[
4 + 5 > 0
\]
\[
9 > 0
\]
Since \(9\) is positive, \(x = 4\) is a valid solution.
**Final Answer:**
\[
x = 4
\]
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución Extra Insights
To solve the equation \( \log_{3}(x+5)=2 \), we can rewrite this in exponential form. That is, \( x + 5 = 3^2 \). Calculating \( 3^2 \) gives us 9, so we have \( x + 5 = 9 \). Now, we can solve for \( x \) by subtracting 5 from both sides: \[ x = 9 - 5 \] Therefore, \( x = 4 \). Isn't it fascinating how logarithmic functions let us translate between multiplication and addition? Just another way math shows its elegant side! If you ever get stuck with logarithms, remember to always convert them to exponential form. A common mistake is to forget this step, leading to an incorrect understanding. So, practice this conversion—it’ll save you from some headaches!
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