Problem 26 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \log _{0,4}\left(\frac{2}{5}\right)^{-3} $$

Step-by-Step Solution

Answer

The simplified expression is approximately -2.756.

Step 1: Simplify the Exponent

The expression is given as \( \log_{0.4}\left( \frac{2}{5} \right)^{-3} \). Start by simplifying the term \( \left( \frac{2}{5} \right)^{-3} \). Recall the rule for negative exponents: \( x^{-n} = \frac{1}{x^n} \). Thus, \( \left( \frac{2}{5} \right)^{-3} = \frac{1}{\left( \frac{2}{5} \right)^3} \).

Step 2: Calculate the Power

Now calculate \( \left( \frac{2}{5} \right)^3 \). This means raising both the numerator and the denominator to the power of 3: \( \frac{2^3}{5^3} = \frac{8}{125} \).

Step 3: Rewrite with Original Base

Substitute back the calculated power into the expression: \( \log_{0.4} \left( \frac{1}{\frac{8}{125}} \right) \). This simplifies to \( \log_{0.4} (\frac{125}{8}) \).

Step 4: Change of Base Formula

Use the change of base formula for logarithms, which states \( \log_b a = \frac{\log_c a}{\log_c b} \). Choose base 10 logarithms: \( \log_{0.4} \left( \frac{125}{8} \right) = \frac{\log_{10} \left( \frac{125}{8} \right)}{\log_{10} 0.4} \).

Step 5: Evaluate the Logarithm

Using a calculator, find \( \log_{10} \left( \frac{125}{8} \right) \approx 1.09691 \) and \( \log_{10} 0.4 \approx -0.39794 \).

Step 6: Simplify the Expression

Finally, evaluate \( \frac{1.09691}{-0.39794} \). This gives approximately \(-2.756\) (rounded to three decimal places).

Key Concepts

Negative Exponents

The concept of negative exponents is fundamental in simplifying expressions. As a general rule, when you see a negative exponent, it indicates a reciprocal. For example, if you have an expression like \( x^{-n} \), it transforms into \( \frac{1}{x^n} \). By using this rule, you can convert negative exponents into positive ones, which are easier to handle in calculations.
In our step-by-step solution, we simplified \( \left( \frac{2}{5} \right)^{-3} \) by applying this rule:\( \left( \frac{2}{5} \right)^{-3} = \frac{1}{\left( \frac{2}{5} \right)^3} \). Once expressed this way, you can proceed with calculating the reciprocal, leading to simpler computations later on.

Change of Base Formula

Logarithms sometimes use different bases, and converting them to a common base can make calculations simpler. This is where the Change of Base Formula comes in handy. The formula is given by \( \log_b a = \frac{\log_c a}{\log_c b} \). This allows you to compute logarithms with any base using a standard calculator, which often operates with base 10 logarithms.
In practice, using the change of base formula can convert a logarithm with any base to one that is more convenient for computations. In our solution, we applied this formula as follows: \( \log_{0.4} \left( \frac{125}{8} \right) = \frac{\log_{10} \left( \frac{125}{8} \right)}{\log_{10} 0.4} \).
By simplifying in this way, you can perform calculations with bases that are easier to evaluate, like base 10, which is often supported by calculators.

Simplifying Fractions

Simplifying fractions is all about reducing them to their simplest form. This means making both the numerator and the denominator as small (but still whole numbers) as possible while maintaining the fraction's value. When dealing with fractions, look for the greatest common divisor (GCD) between the numerator and the denominator.

• Identify the GCD.
• Divide both numerator and denominator by the GCD.

In our problem, after raising \( \frac{2}{5} \) to the power of 3, the fraction became \( \frac{8}{125} \). Another form of simplification we used was rewriting the reciprocal \( \frac{1}{\frac{8}{125}} \) as \( \frac{125}{8} \). Often, simplifying fractions helps in further calculations like when they are used in logarithms or other expressions.
Remember that the ability to simplify fractions is crucial for clearing complex terms and making further mathematical operations easier to manage.