Questions: Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions log(2x+3)−log(x) log(2x+3)−log(x)= (Type an exact answer in simplified form. Use integers or fractions for any numbers in the expression.)

$Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions log(2x+3)−log(x) log(2x+3)−log(x)= (Type an exact answer in simplified form. Use integers or fractions for any numbers in the expression.)$

Transcript text: Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1 . Where possible, evaluate logarithmic expressions \[ \log (2 x+3)-\log (x) \] $\log (2 x+3)-\log (x)=$ $\square$ (Type an exact answer in simplified form. Use integers or fractions for any numbers in the expression.)

Solution

Solution Steps

Solution Approach

To condense the given logarithmic expression, we can use the properties of logarithms. Specifically, we will use the property that states $\log(a) - \log(b) = \log\left(\frac{a}{b}\right)$. Applying this property to the given expression $\log(2x + 3) - \log(x)$, we can combine the two logarithms into a single logarithm.

Step 1: Define the Expression

We start with the logarithmic expression given in the problem: \[ \log(2x + 3) - \log(x) \]

Step 2: Apply the Properties of Logarithms

Using the property of logarithms that states $\log(a) - \log(b) = \log\left(\frac{a}{b}\right)$, we can condense the expression: \[ \log(2x + 3) - \log(x) = \log\left(\frac{2x + 3}{x}\right) \]

Step 3: Simplify the Expression

The condensed expression can be further simplified: \[ \log\left(\frac{2x + 3}{x}\right) = \log\left(2 + \frac{3}{x}\right) \]