Advertisements
Advertisements
प्रश्न
Answer the following:
Without using log tables, prove that `2/5 < log_10 3 < 1/2`
उत्तर
We have to show that, `2/5 < log_10 3 < 1/2`
i.e., to show that,
`2/5 < log_10 3` and `log_10 3 < 1/2`
i.e., to show that,
2 < 5log103 and 2 log103 < 1
i.e., to show that,
2 log1010 < 5 log103 and 2 log103 < log1010 ...[∵ log1010 = 1]
i.e., to show that,
log10102 < log1035 and log1032 < log1010
i.e., to show that,
102 < 35 and 32 < 10
i.e., to show that,
100 < 243 and 9 < 10
which is true
∴ `2/5 < log_10 3 < 1/2`.
APPEARS IN
संबंधित प्रश्न
The function f is defined by \[f\left( x \right) = \begin{cases}x^2 , & 0 \leq x \leq 3 \\ 3x, & 3 \leq x \leq 10\end{cases}\]
The relation g is defined by \[g\left( x \right) = \begin{cases}x^2 , & 0 \leq x \leq 2 \\ 3x, & 2 \leq x \leq 10\end{cases}\]
Show that f is a function and g is not a function.
Let f and g be two real functions defined by \[f\left( x \right) = \sqrt{x + 1}\] and \[g\left( x \right) = \sqrt{9 - x^2}\] . Then, describe function:
(viii) \[\frac{5}{8}\]
Let f(x) = x2 and g(x) = 2x+ 1 be two real functions. Find (f + g) (x), (f − g) (x), (fg) (x) and \[\left( \frac{f}{g} \right) \left( x \right)\] .
If f is a real function satisfying \[f\left( x + \frac{1}{x} \right) = x^2 + \frac{1}{x^2}\]
for all x ∈ R − {0}, then write the expression for f(x).
If f(x) = cos (log x), then the value of f(x2) f(y2) −
The range of the function \[f\left( x \right) = \frac{x^2 - x}{x^2 + 2x}\] is
Check if the relation given by the equation represents y as function of x:
x + y2 = 9
Check if the relation given by the equation represents y as function of x:
3x − 6 = 21
Find x, if g(x) = 0 where g(x) = x3 − 2x2 − 5x + 6
Express the area A of a square as a function of its side s
Check the injectivity and surjectivity of the following function.
f : N → N given by f(x) = x2
Check the injectivity and surjectivity of the following function.
f : Z → Z given by f(x) = x2
Check the injectivity and surjectivity of the following function.
f : R → R given by f(x) = x2
Express the following exponential equation in logarithmic form
25 = 32
Express the following exponential equation in logarithmic form
54° = 1
Prove that logbm a = `1/"m" log_"b""a"`
If f(x) = 3x + 5, g(x) = 6x − 1, then find (fg) (3)
The equation logx2 16 + log2x 64 = 3 has,
Select the correct answer from given alternative.
The domain and range of f(x) = 2 − |x − 5| is
Answer the following:
A function f : R → R defined by f(x) = `(3x)/5 + 2`, x ∈ R. Show that f is one-one and onto. Hence find f–1
Answer the following:
A function f is defined as f(x) = 4x + 5, for – 4 ≤ x < 0. Find the values of f(–1), f(–2), f(0), if they exist
Answer the following:
If b2 = ac. prove that, log a + log c = 2 log b
Answer the following:
If a2 + b2 = 7ab, show that, `log(("a" + "b")/3) = 1/2 log "a" + 1/2 log "b"`
Find the domain of the following function.
f(x) = `sqrtlog(x^2 - 6x + 6)`
Answer the following:
Find the range of the following function.
f(x) = 1 + 2x + 4x
Let X = {3, 4, 6, 8}. Determine whether the relation R = {(x, f(x)) | x ∈ X, f(x) = x2 + 1} is a function from X to N?
A graph representing the function f(x) is given in it is clear that f(9) = 2
Find the following values of the function
(a) f(0)
(b) f(7)
(c) f(2)
(d) f(10)
The data in the adjacent table depicts the length of a person's forehand and their corresponding height. Based on this data, a student finds a relationship between the height (y) and the forehand length (x) as y = ax + b, where a, b are constant.
| Length ‘x’ of forehand (in cm) |
Height 'y' (in inches) |
| 35 | 56 |
| 45 | 65 |
| 50 | 69.5 |
| 55 | 74 |
Check if this relation is a function
The data in the adjacent table depicts the length of a person's forehand and their corresponding height. Based on this data, a student finds a relationship between the height (y) and the forehand length (x) as y = ax + b, where a, b are constant.
| Length ‘x’ of forehand (in cm) |
Height 'y' (in inches) |
| 35 | 56 |
| 45 | 65 |
| 50 | 69.5 |
| 55 | 74 |
Find the height of a person whose forehand length is 40 cm
The domain of the real valued function f(x) = `sqrt((x - 2)/(3 - x))` is ______.
If a function f(x) is given as f(x) = x2 – 6x + 4 for all x ∈ R, then f(–3) = ______.
Let f : R → R be defined by
f(x) = `{(3x; x > 2),(2x^2; 1 ≤ x ≤ 2), (4x; x < 1):}`
Then f(-2) + f(1) + f(3) is ______
Find the domain of the following function.
f(x) = `x/(x^2 + 3x + 2)`
The domain of the function f defined by f(x) = `1/sqrt(x - |x|)` is ______.
Let f and g be two functions given by f = {(2, 4), (5, 6), (8, – 1), (10, – 3)} g = {(2, 5), (7, 1), (8, 4), (10, 13), (11, – 5)} then. Domain of f + g is ______.
Let f(x) = `sqrt(1 + x^2)`, then ______.
Let f(θ) = sin θ (sin θ + sin 3θ) then ______.
