Zadanie 3
Let f(x) = x2 + 2x and g(x) = ex − 6, both with real domain. Find all real x satisfying (f∘g)(x) = 24.
Rational powers and logarithm rules
Manipulate fractional powers and logarithms, change logarithm bases and solve exponential equations with valid domains.
Tę umiejętność sprawdzają 42 zadania wzorowane na arkuszach IB z lat 2022–2026.
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Zadanie 3a
Given log2 p = 4 + log2 r, prove p=16r.
Zadanie 3b
Hence solve log2 p = 4 + log2 r and logr p = 5.
Zadanie 3b
Find k.
Zadanie 3c
When does the reading reach 16°C?
Zadanie 4a
For x > 0, prove log49(16x2) = log7 4 + log7 x.
Zadanie 4b
Hence solve for x > 0.
Zadanie 5a
Show that L6 = ln a and determine the positive integer a.
Zadanie 5b
Hence state L98.
Zadanie 12c
Find N in the form , where a,b,c are integers.
Zadanie 1f
Consider with x ≠ 0, y ≠ 3x/2 and y ≠ −x/2. Under the substitution y = vx, this becomes . Hence show that a solution relation is for some A > 0.
Zadanie 1a
Express log10 56 in terms of p and q.
Zadanie 1b
Express log7 16 in terms of p and q.
Zadanie 1a
Show that (i) f∘2(x) = c1+mxm²; (ii) f∘3(x) = c1+m+m²xm³.
Zadanie 1b
(i) State f∘4(x). (ii) Conjecture a corresponding expression for f∘n(x) for positive integers n. (iii) Use your expression, or another method, to find f∘n(x) when m = 1.
Zadanie 1e
Now let m = −1. (i) Show that f∘n(x) = c/x when n is odd. (ii) Find f∘n(x) when n is even.
Zadanie 4b
How many complete months must pass before the purchasing power of the reserve first exceeds €36 000?
Zadanie 7a
Write 5 as ek for real k.
Zadanie 8a
Show that 1 + log3(2n + 1) ≥ log3(2n + 3) for every positive integer n.
Zadanie 10a
Find f−1(9) as a fraction of integers.
Zadanie 10c
Show that (f∘g)(x) = 729x3.
Zadanie 2b
Hence or otherwise solve log3[x(x2 − 4)] − 1 = log3[(x2 − 11x + 18)(x + 2)].
Zadanie 2a
For the logistic model: (i) interpret choosing L=1000; (ii) find C; (iii) find k; (iv) predict the membership count at t=8, to the nearest whole member.
Zadanie 4
Solve 3log27(6x) − log9 x = 1.
Zadanie 1c
Find the least number n of complete years after which its value is below 15% of its original price.
Zadanie 2a
Find log5(1/a).
Zadanie 2b
Find log125 a.
Zadanie 2
Solve 2 ⋅ 16x + 7 ⋅ 4x − 4 = 0 over the reals.
Zadanie 3b
Find the attenuation score for D.
Zadanie 3c
Another configuration has attenuation score 86. Find its amplitude ratio.
Zadanie 10e
Suppose a=16 and s>0, and define un = An + ln Gn. (i) Show that u_n is arithmetic and find its common difference. (ii) Show that .
Zadanie 12e
Let P(x) = (1 − x)−3/2(1 + x)−3/2. The quadratic approximation to P is P(x) ≈ 1 + 3x2/2. Use x=1/5 in the quadratic approximation 1 + 3x2/2 to obtain a rational approximation to .
Zadanie 1e
Under quota 750, Dn = −1200(9/8)n−1 + 6000. At the end of which processing year does the queue first clear?
Zadanie 1e
The unique contact base lies between 1.05 and 1.1. Find the exact coordinates of the contact point P and the exact base a. You may use the fact that the resulting contact equation has exactly one solution with x>1; an exact candidate must still be verified in…
Zadanie 2e
Find (i) M_3(18); (ii) M_4(18); (iii) M_5(18).
Zadanie 2i
Verify that g(x)=M_x(S) for every admissible integer x.
Zadanie 4
For x≥0, a region is bounded by y = 3x2/(x3 + 4), the x-axis and x=c, where c>0. Its area is ln3. Find c.
Zadanie 8b
Hence show that . You may use the primitive x3(ln x)2/3 − 2x3ln x/9 + 2x3/27.
Zadanie 8b
Hence show that . You may use the primitive x3(ln x)2/3 − 2x3ln x/9 + 2x3/27.
Zadanie 10b
(i) Show that intersection abscissas satisfy x2 − 3dx + 6d = 0. (ii) Hence show that 9d2 − 24d > 0. (iii) Find the possible values of d.
Zadanie 10b
(i) Show that intersection abscissas satisfy x2 − 3dx + 6d = 0. (ii) Hence show that 9d2 − 24d > 0. (iii) Find the possible values of d.
Zadanie 10b
Now suppose the series is arithmetic with common difference d. (i) Show that p=3/4. (ii) Write d as a rational multiple of ln x. (iii) If the sum of its first n terms is , find n.