To express $\log_5(b)$ in terms of a different logarithm base, we can use the change of base formula. The change of base formula allows us to rewrite a logarithm in terms of logarithms of a different base.

Change of Base Formula

For any positive numbers $a$, $b$, and base $c$ (where $c \neq 1$), the change of base formula is:

$\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$

Applying the Formula to $\log_5(b)$

To rewrite $\log_5(b)$ using a different base, say base $10$ (common logarithm) or base $e$ (natural logarithm), we apply the change of base formula:

1. Base 10 (common logarithm):

$\log_5(b) = \frac{\log_{10}(b)}{\log_{10}(5)}$

2. Base $e$ (natural logarithm):

$\log_5(b) = \frac{\ln(b)}{\ln(5)}$

Conclusion

Therefore, $\log_5(b)$ can be expressed as:

$\log_5(b) = \frac{\log_{10}(b)}{\log_{10}(5)} \quad \text{or} \quad \log_5(b) = \frac{\ln(b)}{\ln(5)}.$

These expressions allow you to compute $\log_5(b)$ using either the common logarithm (base 10