To find [tex]\( \log_{3} 5 \)[/tex] using the change-of-base theorem, we can follow these steps:
1. Understand the Change-of-Base Theorem:
The change-of-base theorem states that for any positive numbers [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] (where [tex]\( b \neq 1 \)[/tex] and [tex]\( c \neq 1 \)[/tex]):
[tex]\[
\log_{b} a = \frac{\log_{c}(a)}{\log_{c}(b)}
\][/tex]
Here, [tex]\( b = 3 \)[/tex] and [tex]\( a = 5 \)[/tex]. We can choose any base [tex]\( c \)[/tex] for our logarithms. A common and convenient choice is the natural logarithm (base [tex]\( e \)[/tex]). So we will use natural logarithms ([tex]\( \ln \)[/tex]).
2. Set Up the Calculation:
Using natural logarithms, the change-of-base theorem gives:
[tex]\[
\log_{3} 5 = \frac{\ln(5)}{\ln(3)}
\][/tex]
3. Calculate the Natural Logarithms:
Suppose we have calculated [tex]\( \ln(5) = 1.6094379124341003 \)[/tex] and [tex]\( \ln(3) = 1.0986122886681098 \)[/tex].
4. Divide the Natural Logarithms:
Now, we divide [tex]\( \ln(5) \)[/tex] by [tex]\( \ln(3) \)[/tex]:
[tex]\[
\log_{3} 5 = \frac{1.6094379124341003}{1.0986122886681098} = 1.4649735207179269
\][/tex]
5. Round the Final Answer:
Finally, we round our answer to four decimal places:
[tex]\[
1.4649735207179269 \approx 1.465
\][/tex]
Hence,
[tex]\[
\log_{3} 5 \approx 1.465
\][/tex]
Therefore, [tex]\( \boxed{1.465} \)[/tex] is the logarithm of 5 with base 3, rounded to four decimal places.
