How to Evaluate $3^{\\log_{3}4.52}$ Using Logarithmic Identities

$Math Problem$

Mathematics Grade 10 (Senior High School)

Question Content

Evaluate $3^{\\log_{3}4.52}$

Correct Answer

4.52

Detailed Solution Steps

1
Step 1: Recall the logarithmic identity $a^{\\log_{a}b}=b$, which holds for $a>0, a\\neq1$ and $b>0$.
2
Step 2: In the expression $3^{\\log_{3}4.52}$, we have $a=3$ and $b=4.52$, which satisfy the conditions for the identity.
3
Step 3: Apply the identity directly to get the result: $3^{\\log_{3}4.52}=4.52$.

Knowledge Points Involved

Logarithmic Inverse Identity

The identity $a^{\\log_{a}b}=b$ (where $a>0,a\\neq1,b>0$) describes the inverse relationship between exponential and logarithmic functions. It states that raising a base $a$ to the logarithm of $b$ with base $a$ cancels out the logarithmic and exponential operations, leaving just the argument $b$. This is widely used to simplify exponential expressions with logarithmic exponents.

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