Lessons 12 · 1 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
12
Simple exponential equations and systems
Textbook, Part 1: pp. 69–73
GoalSolve exponential equations by reducing to one base, introducing a new variable and taking out a common factor; solve simple exponential systems.
New words
exponential equation · ko‘rsatkichli tenglamabase of a power · daraja asosiexponent · daraja ko‘rsatkichinew variable · yangi o‘zgaruvchi
Explanation
An equation with the unknown in an exponent is an exponential equation. The key fact: if a > 0 and a ≠ 1, then a^f(x) = a^g(x) is equivalent to f(x) = g(x). The first method reduces both sides to one base using aᵐ · aⁿ = aᵐ⁺ⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 and a⁻ⁿ = 1/aⁿ. In the second method the substitution t = aˣ (t > 0) leads to a quadratic or other algebraic equation; since aˣ is always positive, roots with t ≤ 0 are rejected. The third method takes a common factor out of brackets or divides the equation by one of the powers. In a system of exponential equations we first form a linear system for the exponents or introduce new variables such as u = aˣ, v = bʸ.
9ˣ − 4 · 3ˣ + 3 = 0. Since 9ˣ = (3ˣ)², put t = 3ˣ > 0: t² − 4t + 3 = 0, t = 1 or t = 3. 3ˣ = 1 ⇒ x = 0; 3ˣ = 3 ⇒ x = 1. Answer: x = 0 and x = 1.
Class activity
“Find the base” contest: the teacher gives numbers such as 16, 32, 64, 1/8, 0.25; groups write all of them as powers of 2 as fast as possible (2⁴, 2⁵, 2⁶, 2⁻³, 2⁻²). Then exponential equations are built from these numbers.
Practice
1
For a > 0, a ≠ 1, which equation is a^f(x) = a^g(x) equivalent to?
f(x) = g(x)
2
Solve 2^(x+3) = 32: x = ?
2
3
Solve 3^(2x) − 10 · 3ˣ + 9 = 0.
t = 3ˣ > 0: t² − 10t + 9 = 0, t = 1 or 9 ⇒ x = 0 or x = 2.
4
In 4ˣ + 3 · 2ˣ − 4 = 0 with t = 2ˣ we get t² + 3t − 4 = 0. Why do we reject t = −4? Find x.
2ˣ is positive for every x, so t = −4 is impossible. t = 1 ⇒ 2ˣ = 1 ⇒ x = 0.