What is the value of \( \log_{10} 50 + \frac{\log_{0.5} 5}{1+\log_{2} 5} \)?
Show Hint
- 0
- 1
- 3
- 2
The Correct Option is B
Solution and Explanation
We need to evaluate a numerical expression involving logarithms with different bases.
Step 2: Key Formula or Approach:
We will use the properties of logarithms to simplify the expression. The key properties are: 1. Change of Base Formula: \(\log_a b = \frac{\log_c b}{\log_c a}\) 2. Logarithm of a power: \(\log_a (b^n) = n \log_a b\) 3. Logarithm of a product: \(\log_a (bc) = \log_a b + \log_a c\) 4. Logarithm of a quotient: \(\log_a (\frac{b}{c}) = \log_a b - \log_a c\)
It is often helpful to convert all logarithms to a common base. Let's use base 10 or base 2.
Step 3: Detailed Explanation:
Let's simplify the second term of the expression first: \( \frac{\log_{0.5} 5}{1+\log_{2} 5} \).
First, simplify the numerator, \(\log_{0.5} 5\). \[ \log_{0.5} 5 = \log_{1/2} 5 \] Using the change of base formula to base 2: \[ \log_{1/2} 5 = \frac{\log_2 5}{\log_2 (1/2)} = \frac{\log_2 5}{\log_2 (2^{-1})} = \frac{\log_2 5}{-1} = -\log_2 5 \] Now substitute this back into the fraction: \[ \frac{\log_{0.5} 5}{1+\log_{2} 5} = \frac{-\log_2 5}{1+\log_{2} 5} \] This does not seem to simplify well. Let's try changing all bases to a common base 'c'. \[ \frac{\frac{\log_c 5}{\log_c 0.5}}{1 + \frac{\log_c 5}{\log_c 2}} = \frac{\frac{\log_c 5}{\log_c (1/2)}}{ \frac{\log_c 2 + \log_c 5}{\log_c 2}} = \frac{\frac{\log_c 5}{-\log_c 2}}{ \frac{\log_c (2 \times 5)}{\log_c 2}} = \frac{-\log_c 5 / \log_c 2}{\log_c 10 / \log_c 2} = \frac{-\log_c 5}{\log_c 10} \] Using the change of base formula again, this is equal to \(-\log_{10} 5\).
So the original expression becomes: \[ \log_{10} 50 + (-\log_{10} 5) \] \[ = \log_{10} 50 - \log_{10} 5 \] Using the quotient rule for logarithms: \[ = \log_{10} \left(\frac{50}{5}\right) \] \[ = \log_{10} 10 \] \[ = 1 \] Step 4: Final Answer:
The value of the expression is 1.
Top Questions on Algebra
What is the domain of definition of \( f(x)=\frac{\sqrt{(15-x)(\log_{10}x-1)}}{x-24 \log_{4}2} \)?
- For real numbers \( \alpha \) and \( \beta \), let \( p(x)=x^{2}-(\alpha+\beta)x+\alpha\beta \) and \( q(x)=x^{2}-(\alpha+\beta+2)x+(\alpha+1)(\beta+1) \). Which of the following statements is true?}
- For p,q,r \(\neq\) 0, let \(f(x) = x^3 - px^2 + qx - r\), \(g(x) = x^3 - \frac{p}{r}x^2 + \frac{q}{r^2}x - \frac{1}{r}\). Which of the following statements is/are true?
I. \(f(a) = 0 \implies g(\frac{1}{a}) = 0\).
II. \(f(a) = 0 \implies g(1 + a) = 0\).
III. \(g(a) = 0 \implies f(\frac{r}{a}) = 0\).} - The area enclosed by |x| + |y| = 2 is:
- Suppose M = (1, 1), N = (-1, 3), S = (2, 7), T = (0, -4). If A and B, respectively, divide MN and ST in the ratio 2:3, what is the equation of line AB?
Questions Asked in SRCC GBO exam
Read the given passage and answer the questions.
For those of us without access to outside green space, houseplants are a stylish and affordable way of getting a nature fix. Alongside looking nice, indoor plants actually have several other perks- the biggest benefit of which could be improving your mental health. And the good news is you don’t need to be a self-professed” plant parent” to experience these benefits either. One in eight UK households don’t have access to any kind of garden. Young people and those from minority ethnic backgrounds are among those least likely to have a garden. Not having access to nature can have a number of effects on our health. It’s been linked to symptoms of depression and anxiety, as well as other health conditions, such as asthma, cardiovascular disease, and poor immune function. For many of us, houseplants are an essential link to nature. While there’s not yet a robust body of research on the mental health benefits of houseplants specifically, plenty of studies have shown how beneficial green space and gardening are for mental health. For instance, one study found that people who garden daily have better wellbeing and lower stress levels compared to those who don’t. Gardening also reduces symptoms of depression and anxiety and increases positive emotions to the same extent as biking, walking and eating out. Many of these outcomes are likely to be true of houseplants too. A recent review of 42 studies demonstrates that even just being in the presence of indoor plants can improve mental and physical health. These experiments compared participants doing various activities in rooms with or without plants. The presence of plants saw better performances on cognitive tasks involving focus, sorting or memory recall, greater pain tolerance when holding hands in ice cold water, and lower levels of physiological stress. Interestingly, the aesthetic appearance of plants is important too, with separate research showing that people tend to react more positively to lush, green plants with rounded and denser foliage. But most of these studies are centred around the mere presence of plants. From research on the benefits of gardening, we can assume that caring for houseplants will bring out many more emotional benefits- such as pride, social connection, satisfaction, fascination, mental resilience in times of stress, and may even help you heal from past trauma.... Plants can remove pollutants such as carbon dioxide, nitrogen dioxide (from nearby traffic), fine particles (from dust) and volatile organic compounds (from air fresheners, cooking and cleaning). For people indoors for the majority of the day, indoor air quality is hugely important. High concentrations of carbon dioxide can reduce cognitive performance (such as concentration and memory recall) while prolonged exposure to other indoor pollutants can cause long-term health problems- ranging from minor eye or throat irritations to respiratory problems and cancer. But removing a meaningful quantity of indoor pollutants would require a lot of plants in a very bright room- something unrealistic for most people. If you do want to give it a try, plants with a high leaf area- such as an India rubber tree (Ficus elastic) or devil’s ivy (Epipremnum aureum) are your best bet- SRCC GBO - 2026
- Reading Comprehension
Study the following charts and answer the questions that follow. Chart 1 shows the percentage of revenue generated in four quarters of the year 2020 for a company
- SRCC GBO - 2026
- Pie Charts
The table provided displays the estimated cost (in lakh) for the construction of a canal between two points. Based on the information in the table, answer the questions that follow.
The provided bar graphs outline the population and national income of a country from the fiscal years 2014-15 to 2019-20. For each of the ensuing questions, select the most suitable option.
- SRCC GBO - 2026
- Bar Graph
The following graph shows the revenue generated by three companies A, B and C in different years. Study the graph and answer the questions that follow.
- SRCC GBO - 2026
- Bar Graph