Question

Which expression is equivalent to left-bracket log 9 one-half log x log (x cubed 4) right-bracket minus log 6?

Answer 1

The equivalent expression of $\log(9) + \frac 12 \log(x) + \log(x^3) - \log(6)$ is $3\frac 12\log(\frac 32 x)$

How to determine the equivalent expression?

The logarithmic expression is given as:

$\log(9) + \frac 12 \log(x) + \log(x^3) - \log(6)$

Apply the power rule of logarithm

$\log(9) + \log(x^\frac 12) + \log(x^3) - \log(6)$

Apply the product rule of logarithm

$\log(9 * x^\frac 12 * x^3) - \log(6)$

Evaluate the product

$\log(9 * x^{3\frac 12}) - \log(6)$

Apply the quotient rule of logarithm

$\log(9 * x^{3\frac 12} \div 6)$

Evaluate the quotient

$\log(\frac 32 x^{3\frac 12})$

Rewrite as:

$3\frac 12\log(\frac 32 x)$

Hence, the equivalent expression of $\log(9) + \frac 12 \log(x) + \log(x^3) - \log(6)$ is $3\frac 12\log(\frac 32 x)$

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