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The expression shown is:

\[
^5\log 75 - ^5\log 3 + 1
\]

This uses logarithms with a base of 5. To simplify this expression, we can apply logarithmic properties, such as:

\[
\log_b (x) - \log_b (y) = \log_b \left( \frac{x}{y} \right)
\]

Applying this property:

\[
^5\log 75 - ^5\log 3 = ^5\log \left( \frac{75}{3} \right)
\]

Simplifying \( \frac{75}{3} \):

\[
^5\log 25
\]

Next, using the fact that \( 25 = 5^2 \), we can apply the power rule for logarithms:

\[
^5\log 25 = ^5\log (5^2) = 2 \cdot ^5\log 5
\]

Since \( ^5\log 5 = 1 \) (as the log of a number to its own base is 1), we get:

\[
2 \cdot 1 = 2
\]

Finally, adding the constant 1:

\[
2 + 1 = 3
\]

Thus, the simplified value of the expression is:

\[
\boxed{3}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How can the change of base formula for logarithms be used?
2. What is the power rule for logarithms, and how is it applied?
3. How do logarithmic properties simplify expressions?
4. Can the value of other logarithmic expressions be derived similarly?
5. What are the common logarithmic identities?

**Tip:** Remember, logarithms simplify multiplication, division, and exponentiation into addition, subtraction, and multiplication!

Simplify the logarithmic expression ^5log75 - ^5log3 + 1 using key logarithmic properties. This step-by-step guide covers the subtraction and power rules of logarithms, suitable for students in Grades 9-12.

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