Let's solve the equation [tex]\(\frac{1}{3} \log 27 = \log (0.59)\)[/tex] step-by-step:
1. Understanding the Equation:
The equation compares [tex]\(\frac{1}{3} \log 27\)[/tex] with [tex]\(\log (0.59)\)[/tex]. We need to check if both expressions evaluate to the same value.
2. Simplify the Left Side:
- We start with [tex]\(\frac{1}{3} \log 27\)[/tex]. Recall that [tex]\(\log (a^b) = b \log a\)[/tex].
- Rewrite [tex]\(27\)[/tex] as [tex]\(3^3\)[/tex], so [tex]\(\log 27 = \log (3^3)\)[/tex].
- Using the logarithm power rule, [tex]\(\log (3^3) = 3 \log 3\)[/tex].
- Therefore, [tex]\(\frac{1}{3} \log 27 = \frac{1}{3} \cdot 3 \log 3 = \log 3\)[/tex].
3. Evaluate the Left Side:
- The value of [tex]\(\log 3\)[/tex] is approximately 0.4771.
4. Evaluate the Right Side:
- We need to find the logarithm of 0.59, which is [tex]\(\log (0.59)\)[/tex].
- The value of [tex]\(\log (0.59)\)[/tex] is approximately -0.2291.
5. Comparison:
- We have [tex]\(\frac{1}{3} \log 27\)[/tex] approximately equal to 0.4771.
- We have [tex]\(\log (0.59)\)[/tex] approximately equal to -0.2291.
6. Conclusion:
- The left side [tex]\(\frac{1}{3} \log 27\)[/tex] is 0.4771 and the right side [tex]\(\log (0.59)\)[/tex] is -0.2291.
- Since 0.4771 is not equal to -0.2291, the equation [tex]\(\frac{1}{3} \log 27 = \log (0.59)\)[/tex] does not hold true.
Hence, the given equation [tex]\(\frac{1}{3} \log 27 = \log (0.59)\)[/tex] is not correct, as the two sides do not evaluate to the same number.
