Apply The Change Of Base Formula To Evaluate The Following Expressions:1. $\log_{\frac{1}{5}} 2$2. $\log_{\frac{1}{2}} 3$3. $\log_2 \frac{1}{3}$4. $\log_3 \frac{1}{2}$

Introduction

The change of base formula is a fundamental concept in mathematics, particularly in the field of logarithms. It allows us to express a logarithm in terms of another base, making it easier to evaluate and manipulate logarithmic expressions. In this article, we will apply the change of base formula to evaluate four different logarithmic expressions.

What is the Change of Base Formula?

The change of base formula is given by:

$$\log_b a = \frac{\log_c a}{\log_c b}$$

where $a$, $b$, and $c$ are positive real numbers, and $c \neq 1$. This formula allows us to express a logarithm in terms of another base, $c$.

Applying the Change of Base Formula to Evaluate Logarithmic Expressions

1. Evaluating $\log_{\frac{1}{5}} 2$

To evaluate $\log_{\frac{1}{5}} 2$, we can use the change of base formula with $a = 2$, $b = \frac{1}{5}$, and $c = 10$.

$$\log_{\frac{1}{5}} 2 = \frac{\log_{10} 2}{\log_{10} \frac{1}{5}}$$

Using a calculator, we can evaluate the logarithms:

$$\log_{10} 2 \approx 0.3010$$

$$\log_{10} \frac{1}{5} = -\log_{10} 5 \approx -0.69897$$

Therefore,

$$\log_{\frac{1}{5}} 2 \approx \frac{0.3010}{-0.69897} \approx -0.4304$$

2. Evaluating $\log_{\frac{1}{2}} 3$

To evaluate $\log_{\frac{1}{2}} 3$, we can use the change of base formula with $a = 3$, $b = \frac{1}{2}$, and $c = 10$.

$$\log_{\frac{1}{2}} 3 = \frac{\log_{10} 3}{\log_{10} \frac{1}{2}}$$

Using a calculator, we can evaluate the logarithms:

$$\log_{10} 3 \approx 0.4771$$

$$\log_{10} \frac{1}{2} = -\log_{10} 2 \approx -0.3010$$

Therefore,

$$\log_{\frac{1}{2}} 3 \approx \frac{0.4771}{-0.3010} \approx -1.5857$$

3. Evaluating $\log_2 \frac{1}{3}$

To evaluate $\log_2 \frac{1}{3}$, we can use the change of base formula with $a = \frac{1}{3}$, $b = 2$, and $c = 10$.

$$\log_2 \frac{1}{3} = \frac{\log_{10} \frac{1}{3}}{\log_{10} 2}$$

Using a calculator, we can evaluate the logarithms:

$$\log_{10} \frac{1}{3} = -\log_{10} 3 \approx -0.4771$$

$$\log_{10} 2 \approx 0.3010$$

Therefore,

$$\log_2 \frac{1}{3} \approx \frac{-0.4771}{0.3010} \approx -1.5857$$

4. Evaluating $\log_3 \frac{1}{2}$

To evaluate $\log_3 \frac{1}{2}$, we can use the change of base formula with $a = \frac{1}{2}$, $b = 3$, and $c = 10$.

$$\log_3 \frac{1}{2} = \frac{\log_{10} \frac{1}{2}}{\log_{10} 3}$$

Using a calculator, we can evaluate the logarithms:

$$\log_{10} \frac{1}{2} = -\log_{10} 2 \approx -0.3010$$

$$\log_{10} 3 \approx 0.4771$$

Therefore,

$$\log_3 \frac{1}{2} \approx \frac{-0.3010}{0.4771} \approx -0.6304$$

Conclusion

In this article, we applied the change of base formula to evaluate four different logarithmic expressions. We used the formula to express each logarithm in terms of a common base, $c = 10$, and then evaluated the resulting expressions using a calculator. The results show that the change of base formula is a powerful tool for evaluating logarithmic expressions.

References

• [1] "Change of Base Formula" by Math Open Reference. Retrieved from https://www.mathopenref.com/logarithmchangeofbase.html
• [2] "Logarithms" by Khan Academy. Retrieved from https://www.khanacademy.org/math/precalculus/precalc-logarithms

Additional Resources

• [1] "Logarithmic Functions" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/LogarithmicFunctions.html
• [2] "Change of Base Formula" by Wolfram Alpha. Retrieved from https://www.wolframalpha.com/input/?i=change+of+base+formula