घातांकीय समीकरण समाधानकर्ता
aˣ = b · Linear Exponent · Natural e · Quadratic in eˣ
Solve exponential equations step by step using logarithms. Supports a^x = b, a^(mx+n) = b, natural log forms, and quadratic-in-eˣ equations. Exact + decimal answers with graph.
समीकरण का प्रकार
त्वरित उदाहरण
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चरण-दर-चरण समाधान
ग्राफ़ — प्रतिच्छेद बिंदु
लघुगणक के नियम — त्वरित संदर्भ
घातांकीय समीकरण (Exponential Equations) क्या होते हैं?
An exponential equation is any equation in which the unknown variable appears in an exponent. The simplest form is aˣ = b, asking "to what power must we raise a to get b?" These equations arise constantly in science, finance, and engineering whenever quantities grow or decay multiplicatively over time.
Unlike polynomial equations, exponential equations cannot be solved by standard algebraic manipulation alone. The key tool is the logarithm: since log(aˣ) = x·log(a), taking the log of both sides converts an exponential equation into a linear one that is easy to solve.
घातांकीय समीकरणों को चरण-दर-चरण कैसे हल करते हैं?
Type 1 — Basic Form: aˣ = b
Take the logarithm (any base) of both sides and apply the power rule:
- Take log: log(aˣ) = log(b)
- Power rule: x · log(a) = log(b)
- Isolate x: x = log(b) / log(a) (change-of-base formula)
Example: 5ˣ = 100 → x = log(100)/log(5) = 2/log(5) ≈ 2.861353
Type 2 — Linear Exponent: a^(mx+n) = b
Take log of both sides, expand, then solve the resulting linear equation:
- Take log: (mx + n) · log(a) = log(b)
- Divide: mx + n = log(b)/log(a)
- Subtract n: mx = log(b)/log(a) − n
- Divide by m: x = (log(b)/log(a) − n) / m
Type 3 — Coefficient Form: k·aˣ = b
First divide both sides by k to isolate the exponential term: aˣ = b/k. Then proceed with the basic form. Note: if b/k ≤ 0 there is no real solution.
Type 4 — Natural Exponential: eˣ = c or e^(ax+b) = c
Apply the natural logarithm (ln) since ln and e are inverse operations:
- eˣ = c → x = ln(c)
- e^(ax+b) = c → ax + b = ln(c) → x = (ln(c) − b) / a
Type 5 — Quadratic in eˣ
Equations like e^(2x) − 5eˣ + 6 = 0 become quadratics after substitution u = eˣ:
- Substitute: u² − 5u + 6 = 0
- Factor or quadratic formula: u = 2 or u = 3
- Back-substitute: x = ln(2) ≈ 0.693147 or x = ln(3) ≈ 1.098612
- Discard any u ≤ 0 since eˣ > 0 always
वास्तविक जीवन में घातांकीय समीकरण कहाँ आते हैं?
| Field | Equation | What x represents |
|---|---|---|
| Compound interest | A = P · e^(rt) | Time to reach amount A |
| Radioactive decay | N = N₀ · e^(−λt) | Time until quantity N |
| Population growth | P = P₀ · a^t | Time to reach population P |
| pH chemistry | pH = −log[H⁺] | H⁺ concentration |
| Earthquake (Richter) | M = log(A/A₀) | Amplitude ratio |
| Sound (decibels) | dB = 10 · log(I/I₀) | Intensity ratio |
प्राकृतिक लघुगणक (ln) बनाम सामान्य लघुगणक (log)
Both the natural logarithm (ln, base e) and the common logarithm (log, base 10) produce the same final answer for x because of the change-of-base formula: ln(b)/ln(a) = log(b)/log(a). Use ln when the base is e (it gives the cleanest form), and log for all other bases. Most scientific calculators provide both.
सामान्य गलतियाँ जो टालनी चाहिए
- Forgetting that a must be positive and not equal to 1 for aˣ = b to have a unique solution.
- Taking log of a negative number — since aˣ > 0 always, if b ≤ 0 there is no real solution.
- Dropping the coefficient k in k·aˣ = b without first isolating aˣ.
- Forgetting to discard negative u roots in quadratic-form equations where u = eˣ.
- Using log instead of ln inconsistently — always use the same log throughout one calculation.
- Rounding too early — carry full precision through all steps and round only the final answer.