do the same for $$ \log_5\left(b\right) $$

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

Explore how to express $ \log_5(b) $ using different logarithmic bases with the change of base formula. This detailed explanation covers mathematical concepts related to logarithms and provides step-by-step solutions. Suitable for students in Grades 10-12.

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