Answer :
Let's solve each part of the question step by step:
- Find the logarithm of the following:
a) [tex]\log_2(8)[/tex]
To solve [tex]\log_2(8)[/tex], we need to find the power to which 2 must be raised to get 8. Since [tex]8 = 2^3[/tex], it follows that:
[tex]\log_2(8) = 3[/tex]
b) Simplify: [tex]\log(4) + \log(5)[/tex]
This expression can be simplified using the logarithm property that states [tex]\log_b(m) + \log_b(n) = \log_b(m \times n)[/tex]. Therefore:
[tex]\log(4) + \log(5) = \log(4 \times 5) = \log(20)[/tex]
c) Solve for [tex]x[/tex]: [tex](\log(x))^2 = 5[/tex]
First, take the square root of both sides to find [tex]\log(x)[/tex]:
[tex]\log(x) = \sqrt{5} \quad \text{or} \quad \log(x) = -\sqrt{5}[/tex]
Now, solve for [tex]x[/tex] by rewriting each expression in exponential form:
[tex]x = 10^{\sqrt{5}}[/tex] and [tex]x = 10^{-\sqrt{5}}[/tex]
Thus, [tex]x[/tex] can be either [tex]10^{\sqrt{5}}[/tex] or [tex]10^{-\sqrt{5}}[/tex].
d) Find: [tex]\log_5(125)^4[/tex]
First, calculate [tex]\log_5(125)[/tex]:
Since [tex]125 = 5^3[/tex], it follows that:
[tex]\log_5(125) = 3[/tex]
Now, let's find [tex]\log_5(125)^4[/tex]:
Multiply the result by 4:
[tex](\log_5(125))^4 = (3)^4[/tex]
Therefore, [tex]\log_5(125)^4 = 81[/tex]