Question

1. use transformations of the graph ( f(x)=log _{2} x ) to graph the function ( h(x)=log _{2}(x - 2) )
2. use the properties of logarithms to write each expression as a single logarithm.

( 2 ln (x - 3)-ln (x + 3)-2 ln x )

1. use the properties of logarithms to write each expression as a sum and/or difference of logarithmic expression.

( log left(\frac{x}{sqrt{x^{3} y^{5}}}
ight) )

1. solve for ( x ).

( 2^{x^{2}-7}=2^{6 x} )

Explanation:

Step1: Apply the power rule of logarithms

The power rule states that \(n\ln M=\ln M^{n}\).
For \(2\ln(x - 3)\), we get \(\ln(x - 3)^{2}\); for \(2\ln x\), we get \(\ln x^{2}\).
So the expression \(2\ln(x - 3)-\ln(x + 3)-2\ln x\) becomes \(\ln(x - 3)^{2}-\ln(x + 3)-\ln x^{2}\).

Step2: Apply the quotient rule of logarithms

The quotient rule states that \(\ln M-\ln N=\ln\frac{M}{N}\).
First, combine \(\ln(x - 3)^{2}-\ln(x + 3)\) using the quotient rule: \(\ln\frac{(x - 3)^{2}}{x + 3}\).
Then, subtract \(\ln x^{2}\) (which is equivalent to combining \(\ln\frac{(x - 3)^{2}}{x + 3}\) and \(\ln x^{2}\) using the quotient rule again).
We get \(\ln\frac{(x - 3)^{2}}{x^{2}(x + 3)}\).

Answer:

\(\ln\frac{(x - 3)^{2}}{x^{2}(x + 3)}\)