How do you use the Change of Base Formula and a calculator to evaluate the logarithm #log (1/5)^3#?

Question

Sadie Alexander · Accepted Answer

You can start by rewriting. #=log 5^-3# Then put the exponent in front: #=-3*log5#

Cora Alexander · Answer

undefined To evaluate the logarithm $\log \left(\frac{1}{5}\right)^3$ using the Change of Base Formula and a calculator, you can follow these steps:

1. Apply the exponent rule for logarithms: $\log_b(a^c) = c \cdot \log_b(a)$.
2. Rewrite the logarithm using the exponent rule: $\log \left(\frac{1}{5}\right)^3 = 3 \cdot \log \left(\frac{1}{5}\right)$.
3. Use the Change of Base Formula, which states that $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$, where $c$ is any positive number different from 1.
4. Choose a convenient base for the logarithm. Typically, calculators use base 10 (denoted as $\log_{10}$) or base $e$ (natural logarithm, denoted as $\ln$).
5. Apply the Change of Base Formula: $\log \left(\frac{1}{5}\right) = \frac{\log\left(\frac{1}{5}\right)}{\log(10)}$ or $\log \left(\frac{1}{5}\right) = \frac{\ln\left(\frac{1}{5}\right)}{\ln(10)}$.
6. Use a calculator to evaluate the logarithm using the chosen base.
7. Finally, multiply the result by 3 since we applied an exponent of 3 earlier.

By following these steps, you can accurately evaluate the logarithm $\log \left(\frac{1}{5}\right)^3$ using the Change of Base Formula and a calculator.