Certainly! To evaluate [tex]\(\log (0.75)^3\)[/tex], given that [tex]\(\log 7.5 = 0.875\)[/tex], we can proceed step-by-step as follows:
1. Recognize the Power Rule for Logarithms:
The power rule for logarithms states that [tex]\(\log(a^b) = b \cdot \log(a)\)[/tex]. Therefore, we can express [tex]\(\log(0.75)^3\)[/tex] as:
[tex]\[
\log(0.75)^3 = 3 \cdot \log(0.75)
\][/tex]
2. Express 0.75 in Terms of 7.5:
Notice that [tex]\(0.75\)[/tex] can be rewritten as [tex]\(7.5 / 10\)[/tex]. Hence, we can use the property of logarithms that [tex]\(\log \left( \frac{a}{b} \right) = \log(a) - \log(b)\)[/tex]:
[tex]\[
\log(0.75) = \log\left(\frac{7.5}{10}\right) = \log(7.5) - \log(10)
\][/tex]
3. Use Given Information:
Given that [tex]\(\log 7.5 = 0.875\)[/tex], we substitute this value into the expression. Also, we know that [tex]\(\log 10 = 1\)[/tex] (since it's the common logarithm with base 10):
[tex]\[
\log(0.75) = 0.875 - 1 = -0.125
\][/tex]
4. Apply the Power Rule:
Now, substitute back to find [tex]\(\log(0.75)^3\)[/tex]:
[tex]\[
\log(0.75)^3 = 3 \cdot \log(0.75) = 3 \cdot (-0.125) = -0.375
\][/tex]
Therefore, the value of [tex]\(\log (0.75)^3\)[/tex] is [tex]\(-0.375\)[/tex].
