Let's solve the expression [tex]\(\log_5 75 - \log_5 3\)[/tex] step-by-step.
### Step 1: Recall the Logarithmic Property
One useful property of logarithms is the quotient rule, which states:
[tex]\[
\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)
\][/tex]
In this problem, we are given the expression [tex]\(\log_5 75 - \log_5 3\)[/tex]. We will apply this property.
### Step 2: Apply the Property
Using the quotient rule, we can combine the two logarithms into a single logarithm with the given bases:
[tex]\[
\log_5 75 - \log_5 3 = \log_5\left(\frac{75}{3}\right)
\][/tex]
### Step 3: Simplify the Argument
Next, simplify the argument inside the logarithm:
[tex]\[
\frac{75}{3} = 25
\][/tex]
So, the expression now is:
[tex]\[
\log_5 25
\][/tex]
### Step 4: Evaluate the Logarithm
We need to determine what the exponent is when 5 is raised to some power to obtain 25:
[tex]\[
\log_5 25 = 2
\][/tex]
because [tex]\(5^2 = 25\)[/tex].
### Final Answer
Thus, the value of the original expression [tex]\(\log_5 75 - \log_5 3\)[/tex] is:
[tex]\[
2.0
\][/tex]
As a result, the solution to the problem [tex]\(\log_5 75 - \log_5 3\)[/tex] is [tex]\(2.0\)[/tex].
