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What is the solution of the equation $\log_{a}(x^3) = b$, 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 $\log_{a}(x^3) = b$, where $a$ and $b$ are positive constants?

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

Step 1

Rearranging the Equation

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Answer

To solve the equation, we start by rewriting it using the properties of logarithms. The equation can be expressed as:

x3=abx^3 = a^b

This is derived from the definition of logarithms, where log⁡a(y)=x\log_{a}(y) = x implies y=axy = a^x.

Step 2

Solving for x

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Next, we isolate xx by taking the cube root of both sides. This gives us:

x=(ab)1/3x = (a^b)^{1/3}

Simplifying this expression results in:

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

Step 3

Final Answer

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Answer

Therefore, the solution to the equation is:

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

From the options provided, this corresponds to option B: x=ab/3x = a^{b/3}.

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