Given log 2 = 0.30103, log 3 = 0.477112, find log√18. The first and the eighth terms of an AP are 2 and 23 respectively. Find the tenth term of the series. A code word consists of... Given log 2 = 0.30103, log 3 = 0.477112, find log√18. The first and the eighth terms of an AP are 2 and 23 respectively. Find the tenth term of the series. A code word consists of three letters followed by two digits. How many code words can be made, if neither the letters nor the digits are repeated in any code? If "P5 : P3 = 2: 1", then find the value of n. Solve 5x^2 + 5x - 30 = 0. Find dy/dx, if y = √(x - 1)/√(x + 1). Integrate ∫ e^x^2 dx. 100 students appeared in B.Com examination, 70 secure first class marks in Mathematics and 60 first class marks in Accountancy. If each student secured first class marks in at least one subject, find the number of students who secured first class marks in both subjects.
Understand the Problem
The question presents a set of mathematical problems, primarily involving logarithmic calculations, sequences, coding permutations, and calculus. It requires solutions for various mathematical concepts, which necessitates step-by-step calculations for resolution.
Answer
The value of $\log \sqrt{18}$ is approximately $0.62764$.
Answer for screen readers
The value of $\log \sqrt{18}$ is approximately $0.62764$.
Steps to Solve
- Calculate $\log \sqrt{18}$ Using Logarithmic Properties
The square root can be expressed as an exponent: $$ \sqrt{18} = 18^{1/2} $$
Using the logarithmic property $\log a^b = b \cdot \log a$, we have: $$ \log \sqrt{18} = \log(18^{1/2}) = \frac{1}{2} \log 18 $$
- Express $\log 18$ in Terms of Known Logarithms
To find $\log 18$, we use the fact that $18 = 2 \cdot 9 = 2 \cdot 3^2$: $$ \log 18 = \log(2 \cdot 3^2) = \log 2 + 2 \log 3 $$
Substituting the known values $\log 2 = 0.30103$ and $\log 3 = 0.47712$, we get: $$ \log 18 = 0.30103 + 2(0.47712) $$
- Calculate $\log 18$
Calculating: $$ \log 18 = 0.30103 + 0.95424 = 1.25527 $$
- Substitute Back to Find $\log \sqrt{18}$
Now substitute $\log 18$ back into the expression for $\log \sqrt{18}$: $$ \log \sqrt{18} = \frac{1}{2} \cdot 1.25527 = 0.627635 $$
The value of $\log \sqrt{18}$ is approximately $0.62764$.
More Information
Logarithmic properties are extremely useful in breaking down complex logarithmic expressions into simpler components. By using the known logarithm values, we can compute the logarithm of other numbers systematically.
Tips
- Forgetting logarithmic identities: Always remember to use properties like $\log a^b = b \cdot \log a$ and the product rule $\log(ab) = \log a + \log b$.
- Calculating square roots incorrectly: Make sure to convert square roots correctly into their exponent form.
