Use the properties of logarithms to expand [tex]\log \left(x^3 z\right)[/tex]. Each logarithm should involve only one variable and should not have any exponents or fractions. Assume that all variables are positive.
To expand [tex]\(\log \left(x^3 z\right)\)[/tex] using the properties of logarithms, follow these steps:
1. Apply the Product Rule of Logarithms: The logarithm of a product is the sum of the logarithms of the individual factors. Specifically, [tex]\(\log(ab) = \log(a) + \log(b)\)[/tex].
Applying this to [tex]\(\log(x^3 z)\)[/tex]: [tex]\[
\log(x^3 z) = \log(x^3) + \log(z)
\][/tex]
2. Apply the Power Rule of Logarithms: The logarithm of a power is the exponent multiplied by the logarithm of the base. Specifically, [tex]\(\log(a^b) = b \log(a)\)[/tex].
Applying this to [tex]\(\log(x^3)\)[/tex]: [tex]\[
\log(x^3) = 3 \log(x)
\][/tex]
3. Combine the Results: Substitute the expanded form of [tex]\(\log(x^3)\)[/tex] back into the expression: [tex]\[
\log(x^3 z) = 3 \log(x) + \log(z)
\][/tex]
So, the expanded form of [tex]\(\log \left(x^3 z\right)\)[/tex] is: [tex]\[
\log \left(x^3 z\right) = 3 \log(x) + \log(z)
\][/tex]
