Advertisements
Advertisements
Question
Write the following expression as a single logarithm.
ln (x + 2) + ln (x − 2) − 3 ln (x + 5)
Solution
ln (x + 2) + ln (x − 2) − 3 ln (x + 5)
= ln [(x + 2) (x – 2)] – ln (x + 5)3 ...`[(log"m" + log"n" = log"mn"),("n" log "m" = log "m"^"n")]`
= ln(x2 − 4) − ln(x + 5)3
= In `((x^2 - 4)/(x + 5)^3) ...[log "m" - log "n" = log "m"/"n"]`
APPEARS IN
RELATED QUESTIONS
Let f : R+ → R, where R+ is the set of all positive real numbers, such that f(x) = loge x. Determine
(a) the image set of the domain of f
Let X = {1, 2, 3, 4} and Y = {1, 5, 9, 11, 15, 16}
Determine which of the set are functions from X to Y.
(b) f2 = {(1, 1), (2, 7), (3, 5)}
Let A = [p, q, r, s] and B = [1, 2, 3]. Which of the following relations from A to B is not a function?
If f(x) = (x − a)2 (x − b)2, find f(a + b).
If f(x) = (a − xn)1/n, a > 0 and n ∈ N, then prove that f(f(x)) = x for all x.
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:
(iv) \[\frac{f}{g}\]
If f, g and h are real functions defined by
If\[f\left( x \right) = 1 - \frac{1}{x}\] , then write the value of \[f\left( f\left( \frac{1}{x} \right) \right)\]
Write the domain and range of the function \[f\left( x \right) = \frac{x - 2}{2 - x}\] .
Write the domain and range of function f(x) given by
Write the domain and range of function f(x) given by \[f\left( x \right) = \sqrt{\left[ x \right] - x}\] .
Which one of the following is not a function?
If f(x) = cos (log x), then value of \[f\left( x \right) f\left( 4 \right) - \frac{1}{2} \left\{ f\left( \frac{x}{4} \right) + f\left( 4x \right) \right\}\] is
If 2f (x) − \[3f\left( \frac{1}{x} \right) = x^2\] (x ≠ 0), then f(2) is equal to
The function f : R → R is defined by f(x) = cos2 x + sin4 x. Then, f(R) =
If f : R → R and g : R → R are defined by f(x) = 2x + 3 and g(x) = x2 + 7, then the values of x such that g(f(x)) = 8 are
If \[e^{f\left( x \right)} = \frac{10 + x}{10 - x}\] , x ∈ (−10, 10) and \[f\left( x \right) = kf\left( \frac{200 x}{100 + x^2} \right)\] , then k =
The range of the function f(x) = |x − 1| is
Which of the following relations are functions? If it is a function determine its domain and range:
{(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}
A function f is defined as follows: f(x) = 5 − x for 0 ≤ x ≤ 4. Find the value of x such that f(x) = 3
Check the injectivity and surjectivity of the following function.
f : R → R given by f(x) = x3
lf f(x) = 3(4x+1), find f(– 3)
Express the following exponential equation in logarithmic form
3–4 = `1/81`
Given that log 2 = a and log 3 = b, write `log sqrt(96)` in terms of a and b
Prove that `"b"^(log_"b""a"` = a
Prove that logbm a = `1/"m" log_"b""a"`
If f(x) = 3x + 5, g(x) = 6x − 1, then find (fg) (3)
Answer the following:
Identify the following relation is the function? If it is a function determine its domain and range.
{(0, 0), (1, 1), (1, –1), (4, 2), (4, –2), (9, 3), (9, –3), (16, 4), (16, –4)}
Answer the following:
Find value of `(3 + log_10 343)/(2 + 1/2 log_10 (49/4) + 1/2 log_10 (1/25)`
Answer the following:
Find the domain of the following function.
f(x) = `(x^2 + 4x + 4)/(x^2 + x - 6)`
Answer the following:
Find the domain of the following function.
f(x) = x!
Let f = {(x, y) | x, y ∈ N and y = 2x} be a relation on N. Find the domain, co-domain and range. Is this relation a function?
Given the function f: x → x2 – 5x + 6, evaluate f(2)
A plane is flying at a speed of 500 km per hour. Express the distance ‘d’ travelled by the plane as function of time t in hour
The domain of the function f(x) = log3+x (x2 - 1) is ______.
Find the range of the following functions given by `sqrt(16 - x^2)`
If f(x) = y = `(ax - b)/(cx - a)`, then prove that f(y) = x.
If f(x) = `log_e{((1 - x))/((1 - x))}, |x| < 1, f{(2x)/((1 + x^2))}` is equal to ______.
The range of the function f(x) = `""^(7 - x)P_(x - 3)` is ______.
Range of the function f(x) = `x/(1 + x^2)` is ______.
