Solve for x log base x of 512=3/2

Problem

(log_x)(512)=3/2

Solution

1. Rewrite the logarithmic equation in its equivalent exponential form using the definition (log_b)(a)=c⇔bc=a

x(3/2)=512

2. Isolate the variable x by raising both sides of the equation to the reciprocal power of the current exponent, which is 2/3

(x(3/2))(2/3)=512(2/3)

3. Simplify the left side using the power of a power rule (am)n=a(m⋅n)

x=512(2/3)

4. Evaluate the right side by first taking the cube root of 512 and then squaring the result.

x=(√(3,512))2

5. Calculate the cube root of 512 which is 8 because 8=512

x=8

6. Square the value to find the final result.

x=64

Final Answer

(log_x)(512)=3/2⇒x=64

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