Subtraction of Logarithms: Applying the formula

To solve the problem, we employ the property of logarithms for subtraction:

• Step 1: Recognize the expression $\log_5 3 - \log_5 2$.
• Step 2: Apply the logarithmic property for subtraction, $\log_b a - \log_b c = \log_b \left( \frac{a}{c} \right)$.
• Step 3: Substitute into the property: $\log_5 3 - \log_5 2 = \log_5 \left( \frac{3}{2} \right)$.

By applying the property, we simplify the expression to $\log_5 \frac{3}{2}$. This is equivalent to $\log_5 1.5$. Therefore:

Therefore, the result of the expression is $\log_5 1.5$.

To solve the problem of evaluating $\log_2 9 - \log_2 3$, we apply the properties of logarithms as follows:

• Step 1: Recognize that the expression uses a subtraction of logarithms with the same base: $\log_2 9 - \log_2 3$.
• Step 2: Use the logarithmic subtraction rule: $\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)$.
• Step 3: Simplify using this rule: $\log_2 9 - \log_2 3 = \log_2 \left(\frac{9}{3}\right)$.
• Step 4: Perform the division: $\frac{9}{3} = 3$.
• Step 5: Therefore, $\log_2 \left(\frac{9}{3}\right) = \log_2 3$.

Thus, the simplified and evaluated result is $\log_2 3$.

To solve the problem, let's use the rules of logarithms:

• Step 1: Recognize that we are dealing with the subtraction of logarithms sharing the same base, which calls for the identity $\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)$.
• Step 2: Apply this identity to the expression $\log_7 5 - \log_7 2$.
• Step 3: Realize that this can thus be expressed as a single logarithm: $\log_7 \left(\frac{5}{2}\right)$.
• Step 4: Simplify the fraction, yielding $\log_7 2.5$.

Therefore, the simplification results in the expression: $\log_7 2.5$.

This matches the correct answer from the given choices.

To solve the problem $\frac{1}{2}\log_39-\log_31.5$, we need to apply the rules of logarithms:

• **Step 1: Simplify with the power rule**
  Using the power rule $\frac{1}{2}\log_39 = \log_39^{1/2}$. Since $9 = 3^2$, we have $9^{1/2} = 3^{2 \cdot 1/2} = 3^1$. Thus, $\frac{1}{2}\log_39 = \log_3 3 = 1$.
• **Step 2: Apply the subtraction rule**
  Now, the expression becomes $1 - \log_3 1.5$. Using the subtraction rule: $1 - \log_3 1.5 = \log_3 3 - \log_3 1.5 = \log_3 \left(\frac{3}{1.5}\right)$.
• **Step 3: Simplify the fraction**
  Calculate $\frac{3}{1.5}$: it simplifies to 2 because $3 \div 1.5 = 2$.

Thus, the simplified expression is $\log_3 2$.

Using the provided answer choices, the correct answer matches choice $\log_3 2$, which corresponds to choice 2.

Therefore, the solution to the problem is $\log_3 2$.

We break it down into parts

$\log_61296=x$

$6^x=1296$

$x=4$

$\frac{1}{4}\cdot4\cdot\log_6\frac{1}{2}-\log_63=$

$\log_6\frac{1}{2}-\log_63=$

$\log_6\left(\frac{1}{2}:3\right)=\log_6\frac{1}{6}$

$\log_6\frac{1}{6}=x$

$6^x=\frac{1}{6}$

$x=-1$