Answer :
Let's simplify each of the logarithmic expressions step by step using logarithm powers and properties.
(i) [tex]\log_2 8^3[/tex]
Step 1: Use the power rule of logarithms, which states [tex]\log_b a^n = n \cdot \log_b a[/tex]. This means you can bring the exponent down as a coefficient.
[tex]\log_2 8^3 = 3 \cdot \log_2 8[/tex]
Step 2: Simplify [tex]\log_2 8[/tex]. Since 8 is a power of 2 (specifically, [tex]2^3[/tex]), we can rewrite it as:
[tex]\log_2 8 = \log_2 (2^3) = 3[/tex]
Step 3: Substitute back:
[tex]3 \cdot \log_2 8 = 3 \cdot 3 = 9[/tex]
Thus, [tex]\log_2 8^3 = 9[/tex].
(ii) [tex]4 \cdot \log_9 3[/tex]
In some problems, there might be specific properties or numbers involved to simplify, but in this case, there is no such simplification possible directly since no powers relate directly.
Thus, [tex]4 \cdot \log_9 3[/tex] stays as is unless more context or restrictions are applied.
(iii) [tex]\log_5 \frac{1}{125}[/tex]
Step 1: Use the property of logarithms that states [tex]\log_b \frac{1}{a} = -\log_b a[/tex]. First, rewrite the expression:
[tex]\log_5 \frac{1}{125} = -\log_5(125)[/tex]
Step 2: Simplify [tex]\log_5 125[/tex]. Since 125 is a power of 5 (specifically, [tex]5^3[/tex]), rewrite it using that power:
[tex]\log_5(125) = \log_5(5^3) = 3[/tex]
Step 3: Substitute back:
[tex]-\log_5 125 = -3[/tex]
Thus, [tex]\log_5 \frac{1}{125} = -3[/tex].
So, the final simplified forms are:
- [tex]\log_2 8^3 = 9[/tex]
- [tex]4 \cdot \log_9 3[/tex] (remains unchanged without further context)
- [tex]\log_5 \frac{1}{125} = -3[/tex]
These calculations use fundamental properties of logarithms to simplify each expression.