Expand your knowledge base with the help of IDNLearn.com's extensive answer archive. Our platform is designed to provide trustworthy and thorough answers to any questions you may have.

Use the change of base formula to evaluate the logarithms to the nearest thousandth.

Change of base formula:
[tex]\[ \log _a x=\frac{\log _b x}{\log _b a} \][/tex]

Evaluate the following:

[tex]\[
\begin{array}{l}
\log _3 6=\square \\
\log _5 20=\square \\
\log _2\left(\frac{1}{5}\right)=\square
\end{array}
\][/tex]


Sagot :

To evaluate the given logarithms, we will use the change of base formula:

[tex]\[ \log_a x = \frac{\log_b x}{\log_b a} \][/tex]

where [tex]\( \log_b \)[/tex] denotes the logarithm with base [tex]\( b \)[/tex]. We will use the natural logarithm (base [tex]\( e \)[/tex]) for convenience.

### 1. Evaluating [tex]\( \log_3 6 \)[/tex]:

Using the change of base formula, we get:

[tex]\[ \log_3 6 = \frac{\log 6}{\log 3} \][/tex]

Note that [tex]\( \log \)[/tex] denotes the natural logarithm here.

The result of calculating [tex]\( \log_3 6 \)[/tex] rounded to the nearest thousandth is:

[tex]\[ \log_3 6 \approx 1.631 \][/tex]

### 2. Evaluating [tex]\( \log_5 20 \)[/tex]:

Similarly, using the change of base formula, we get:

[tex]\[ \log_5 20 = \frac{\log 20}{\log 5} \][/tex]

The result of calculating [tex]\( \log_5 20 \)[/tex] rounded to the nearest thousandth is:

[tex]\[ \log_5 20 \approx 1.861 \][/tex]

### 3. Evaluating [tex]\( \log_2 \left( \frac{1}{5} \right) \)[/tex]:

Again, using the change of base formula, we have:

[tex]\[ \log_2 \left( \frac{1}{5} \right) = \frac{\log \left( \frac{1}{5} \right)}{\log 2} \][/tex]

Since [tex]\( \frac{1}{5} \)[/tex] is a fraction, its logarithm is negative. The result of calculating [tex]\( \log_2 \left( \frac{1}{5} \right) \)[/tex] rounded to the nearest thousandth is:

[tex]\[ \log_2 \left( \frac{1}{5} \right) \approx -2.322 \][/tex]

Summarizing, the evaluated logarithms are:
[tex]\[ \begin{array}{l} \log_3 6 \approx 1.631 \\ \log_5 20 \approx 1.861 \\ \log_2 \left( \frac{1}{5} \right) \approx -2.322 \end{array} \][/tex]