Certainly! Let’s find the value of [tex]\( \log_5(2) \)[/tex] step-by-step.
### Steps to Solve [tex]\( \log_5(2) \)[/tex]
1. Understand the Problem:
- The problem requires finding the logarithm of 2 with base 5, denoted as [tex]\( \log_5(2) \)[/tex]. This tells us the power to which 5 needs to be raised to yield 2.
2. Use the Change of Base Formula:
- The change of base formula for logarithms states that:
[tex]\[
\log_b(a) = \frac{\log_c(a)}{\log_c(b)}
\][/tex]
- Here, [tex]\( a = 2 \)[/tex], [tex]\( b = 5 \)[/tex], and we can use any common logarithm base [tex]\( c \)[/tex] such as 10 (common logarithm [tex]\(\log\)[/tex]) or [tex]\( e \)[/tex] (natural logarithm [tex]\(\ln\)[/tex]). For convenience, we'll use the natural logarithm [tex]\( \ln \)[/tex], but the common logarithm would work equally well.
3. Apply the Formula:
- Plugging in the values, we get:
[tex]\[
\log_5(2) = \frac{\ln(2)}{\ln(5)}
\][/tex]
4. Evaluate the Natural Logarithms:
- Using a calculator or mathematical tables, find the natural logarithms of 2 and 5:
[tex]\[
\ln(2) \approx 0.693147 \quad \text{and} \quad \ln(5) \approx 1.609438
\][/tex]
5. Compute the Quotient:
- Next, divide the natural logarithm of 2 by the natural logarithm of 5:
[tex]\[
\log_5(2) = \frac{0.693147}{1.609438} \approx 0.430677
\][/tex]
Therefore, the value of [tex]\( \log_5(2) \)[/tex] is approximately [tex]\( 0.43067655807339306 \)[/tex].
