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What is the solution of the equation $ ext{log}_a x^3 = b$, where $a$ and $b$ are positive constants? - HSC - SSCE Mathematics Advanced - Question 8 - 2023 - Paper 1

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What is the solution of the equation $ ext{log}_a x^3 = b$, where $a$ and $b$ are positive constants?

Worked Solution & Example Answer:What is the solution of the equation $ ext{log}_a x^3 = b$, where $a$ and $b$ are positive constants? - HSC - SSCE Mathematics Advanced - Question 8 - 2023 - Paper 1

Step 1

Identify the equation

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Answer

The given equation is extlogax3=b ext{log}_a x^3 = b.

Step 2

Rewrite using the definition of logarithm

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Answer

Using the definition of logarithms, we can rewrite this equation as: x3=ab.x^3 = a^b. This states that x3x^3 is equal to aa raised to the power of bb.

Step 3

Solve for x

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Answer

To find xx, we take the cube root of both sides: x=(ab)1/3.x = (a^b)^{1/3}. By applying the property of exponents, this can be simplified to: x=ab/3.x = a^{b/3}.

Step 4

Finalize the answer

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Answer

Thus, the solution to the equation is:

B. x=ab/3x = a^{b/3}

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