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Express the given expression without logs, in simplest form. Assume all variables represent positive values.

(3^(log_(3)(4x^(3))-log_(3)(7w)))
Answer:

Express the given expression without logs, in simplest form. Assume all variables represent positive values. $\newline$ $\left(3^{\log _{3}\left(4 x^{3}\right)-\log _{3}(7 w)}\right)$ $\newline$ Answer:

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Multiplication with rational exponents

Full solution

Q. Express the given expression without logs, in simplest form. Assume all variables represent positive values.

\newline

\left(3^{\log _{3}\left(4 x^{3}\right)-\log _{3}(7 w)}\right)

\newline

Answer:

Apply Quotient Rule: We are given the expression $3^{\log_{3}(4x^{3})-\log_{3}(7w)}$ . To simplify this expression, we will use the properties of logarithms to combine the logs into a single log expression.
Combine Logs: The property of logarithms that we will use is the quotient rule, which states that $\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)$ . Applying this to our expression, we get: $\newline$ $\log_{3}(4x^{3}) - \log_{3}(7w) = \log_{3}\left(\frac{4x^{3}}{7w}\right)$
Remove Log and Exponent: Now we have a single logarithm expression: $\log_{3}\left(\frac{4x^{3}}{7w}\right)$ . The next step is to use the property that $b^{\log_b(m)} = m$ , where $b$ is the base of the logarithm and $m$ is the argument of the logarithm. This means we can remove the logarithm and the base $3$ exponent from our expression, leaving us with the argument of the logarithm. $\newline$ $3^{\log_{3}\left(\frac{4x^{3}}{7w}\right)} = \frac{4x^{3}}{7w}$
Simplify Expression: We have now expressed the original expression without logs. The expression in its simplest form is: $\newline$ egin{equation} $\newline$ \frac{ $4$ x^{ $3$ }}{ $7$ w} $\newline$ egin{equation}

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