Find \(f^{-1}(x)\) for \(f(x)=a^x\)

📝 Question

Let:

\[ f:\mathbb{R}\to \mathbb{R}^+, \quad f(x)=a^x,\quad a>0,\ a\ne1 \]

Find \(f^{-1}(x)\).

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✅ Solution

🔹 Step 1: Check invertibility

The function \(a^x\) is strictly monotonic (increasing if \(a>1\), decreasing if \(0

Hence, it is one-one and onto \(\mathbb{R}^+\).

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🔹 Step 2: Find inverse

Let:

\[ y=a^x \]

Take logarithm base \(a\):

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Interchanging \(x\) and \(y\):

\[ f^{-1}(x)=\log_a x \] —

🎯 Final Answer

\[ \boxed{f^{-1}(x)=\log_a x} \]

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🚀 Exam Shortcut

• Exponential ⇄ Logarithm
• \(a^x \leftrightarrow \log_a x\)
• Always swap variables after solving