Alright, let's evaluate the given logarithms step-by-step.
### 1. Calculating [tex]\(\log_{12} 144\)[/tex]
We are asked to find [tex]\(\log_{12} 144\)[/tex]. The logarithm [tex]\(\log_b x\)[/tex] asks the question: "To what power must we raise [tex]\(b\)[/tex] to get [tex]\(x\)[/tex]?
Here, we need to find the power [tex]\(a\)[/tex] such that:
[tex]\[ 12^a = 144 \][/tex]
By calculation, we find that:
[tex]\[ 12^2 = 144 \][/tex]
Thus:
[tex]\[ \log_{12} 144 = 2 \][/tex]
So, [tex]\(\log_{12} 144 = 2.0\)[/tex].
### 2. Calculating [tex]\(\log_{15} 1\)[/tex]
We are asked to find [tex]\(\log_{15} 1\)[/tex]. The logarithm [tex]\(\log_b x\)[/tex] in this case is asking for:
[tex]\[ b^a = x \][/tex]
Here, we need to find the power [tex]\(a\)[/tex] such that:
[tex]\[ 15^a = 1 \][/tex]
By definition, any number to the power of 0 is 1:
[tex]\[ 15^0 = 1 \][/tex]
Thus:
[tex]\[ \log_{15} 1 = 0 \][/tex]
So, [tex]\(\log_{15} 1 = 0.0\)[/tex].
### 3. Calculating [tex]\(\log_3 \left(\frac{1}{81}\right)\)[/tex]
We are asked to find [tex]\(\log_3 \left(\frac{1}{81}\right)\)[/tex]. This logarithm is asking for:
[tex]\[ 3^a = \frac{1}{81} \][/tex]
We know that:
[tex]\[ 81 = 3^4 \][/tex]
Thus:
[tex]\[ \frac{1}{81} = 3^{-4} \][/tex]
Therefore:
[tex]\[ 3^a = 3^{-4} \][/tex]
[tex]\[ a = -4 \][/tex]
So:
[tex]\[ \log_3 \left(\frac{1}{81}\right) = -4 \][/tex]
Thus, [tex]\(\log_3 \left(\frac{1}{81}\right) = -4.0\)[/tex].
### 4. Calculating [tex]\(\log 0.00001\)[/tex]
In this case, we are finding the common logarithm (base 10) of [tex]\(0.00001\)[/tex]. The logarithm [tex]\(\log_b x\)[/tex] in this instance is asking for:
[tex]\[ 10^a = 0.00001 \][/tex]
We know that:
[tex]\[ 0.00001 = 10^{-5} \][/tex]
Therefore:
[tex]\[ 10^a = 10^{-5} \][/tex]
[tex]\[ a = -5 \][/tex]
So:
[tex]\[ \log 0.00001 = -5 \][/tex]
Thus, [tex]\(\log 0.00001 = -5.0\)[/tex].
To summarize:
[tex]\[
\log_{12} 144 = 2.0
\][/tex]
[tex]\[
\log_{15} 1 = 0.0
\][/tex]
[tex]\[
\log_3 \left(\frac{1}{81}\right) = -4.0
\][/tex]
[tex]\[
\log 0.00001 = -5.0
\][/tex]
These are the evaluated logarithms.
