0
If x = log, (ab), y = loga (bc), z = log (ca),
then which of the following is correct?
Answers
Answer:
Solution:
Concept:
If x = loga b then x =
log (mn) = log m + log n
Calculations:
Given, x = logc (ab), y = loga (bc), z = logb (ca),
Consider,x = logc (ab)
By logarithm property, we have
, and
⇒ from this and
Consider,
⇒1 + x = 1 +
By usiing logarithm property log (mn) = log m + log n, we have
⇒1 + x =
⇒ 1+ x =
Similarly, 1 + y = and 1 + z =
Consider,
=
=
= 1
Share with friends:
Related Exams
NDACoCubesL&TAMCATTech MahindraElitmusTCSWiproHPHCLDeloitteInfosys
Related Questions
Q1. If the function f given byf(x) = x3 – 3(a – 2) x2 + 3ax + 7 for some a ∈ R is increasing in (0, 1] and decreasing in [1, 5), then a root of the equation, \(\frac{{{\rm{f}}\left( {\rm{x}} \right) - 14}}{{{{({\rm{x}} - 1)}^2}}} = 0\left( {{\rm{x}} \ne 1} \right)\)is
Q2. Let f(x) = x2, x ∈ R. For any A ⊆ R, define g(A) = {x ∈ R : f(x) ∈ A}. If S = [0, 4], then which one of the following statements is not true?
Q3. Let f : R → R be a differentiable function satisfying f'(3) + f'(2) = 0. Then \(\mathop {{\rm{lim}}}\limits_{x \to 0} {\left( {\frac{{1 + f\left( {3 + x} \right) - f\left( 3 \right)}}{{1 + f\left( {2 - x} \right) - f\left( 2 \right)}}} \right)^{\frac{1}{x}}}\)is equal to:
Q4. Let f : [-1, 3] → R be defined as \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\left| x \right| + \left[ x \right],}&{ - 1 \le x < 1}\\ {x + \left| x \right|,}&{1 \le x < 2}\\ {x + \left[ x \right],}&{2 \le x \le 3} \end{array}} \right.\) where [t] denotes the greatest integer less than or equal to it. Then, f is discontinuous at:
Q5. Let \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {{\rm{max}}\left\{ {\left| x \right|,{x^2}} \right\},}&{\left| x \right| \le 2}\\ {8 - 2\left| x \right|,}&{2 < \left| x \right| \le 4} \end{array}} \right.\) Let S be the set of points in the interval (-4, 4) at which f is not differentiable. Then S:
Q6. The domain of the definition of the function \(f\left( x \right) = \frac{1}{{4 - {x^2}}} + lo{g_{10}}\left( {{x^3} - x} \right)\) is:
Q7. Let a function f : (0, ∞) → (0, ∞) be defined by \(f\left( x \right) = \left| {1 - \frac{1}{x}} \right|\). Then f is:
Q8. If \({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{{\rm{sin}}\left( {{\rm{p}} + 1} \right){\rm{x}} + {\rm{sinx}}}}{{\rm{x}}},}&{{\rm{x}} < 0}\\ {q,}&{x = 0}\\ {\frac{{\sqrt {{\rm{x}} + {{\rm{x}}^2}} - \sqrt {\rm{x}} }}{{{{\rm{x}}^{3/2}}}},}&{x > 0} \end{array}} \right.\)is continuous at x = 0, then the ordered pair (p, q) is equal to:
Q9. Let f : R → R be defined by \(f\left( x \right)=\frac{x}{1+{{x}^{2}}}\), x ∈ R. Then the range of f is:
Q10. The maximum value of the function f(x) = 3x3 – 18x2 + 27x – 40 on the set S = {x ∈ R : x2 + 30 ≤ 11x} is:
Q11. Let \({{f}_{\text{k}}}\left( x \right)=\frac{1}{\text{k}}\left( \text{si}{{\text{n}}^{\text{k}}}x+\text{co}{{\text{s}}^{\text{k}}}x \right)\) for k = 1, 2, 3, … Then for all x ∈ R, the value of f4(x) – f6(x) is equal to:
Q12. If \(f\left( x \right) = lo{g_e}\left( {\frac{{1 - x}}{{1 + x}}} \right)\), |x| < 1, then \(f\left( {\frac{{2x}}{{1 + {x^2}}}} \right)\) is equal to:
Q13. Let A = {x ∈ R : x is not a positive integer}. Define a function \(f:A \to R{\rm{\;as\;}}f\left( x \right) = \frac{{2x}}{{x - 1}}\) then f is:
Q14. For x ∈ R-{0, 1}. Let \({{\text{f}}_{1}}\left( \text{x} \right)=\frac{1}{\text{x}},\text{ }\!\!~\!\!\text{ }{{\text{f}}_{2}}\left( \text{x} \right)=1-\text{x }\!\!~\!\!\text{ and }\!\!~\!\!\text{ }{{\text{f}}_{3}}\left( \text{x} \right)=\frac{1}{1-\text{x}}\) be three given functions. If a function, J(x) satisfies (f2∘J∘f1)(x) = f3(x) then J(x) is equal to:
Q15. Let A, B, and C be sets such that ϕ ≠ A ∩ B ⊆ C. Then which of the following statements is not true?