Evaluate 9^( log base 3 of 20)

Problem

9^(log_3)(20)

Solution

1. Identify the base of the exponent and the base of the logarithm. The expression is in the form a^(log_b)(c) where a=9 b=3 and c=20

2. Rewrite the base of the exponent to match the base of the logarithm. Since 9=3^2 substitute this into the expression.

(3^2)^(log_3)(20)

3. Apply the power of a power rule for exponents, which states (x^m)^n=x^(m*n)

3^(2⋅(log_3)(20))

4. Apply the power property of logarithms, n⋅(log_b)(x)=(log_b)(x^n) to move the coefficient into the exponent of the argument.

3^(log_3)(20^2)

5. Use the inverse property of logarithms and exponents, b^(log_b)(x)=x to simplify the expression.

20^2

6. Calculate the final value by squaring the number.

400

Final Answer

9^(log_3)(20)=400

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