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ALGEBRALogarithmsMathematics Calculator

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Antilog — Inverse of Logarithm

antilog_b(y) = b^y. Convert pH to concentration, decibels to power ratio, Richter to amplitude. Step-by-step solutions and charts.

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MATHEMATICSLogarithms

Antilogarithm — Reversing the Log

Compute antilog_b(y) = b^y. Convert pH to [H⁺], decibels to ratios, Richter to amplitude. Exponential growth curves and base comparison.

Logarithm →Natural Log (ln) →

📐 Quick Examples — Click to Load

Enter Base and Logarithm Value

Compute $\text{antilog}_b(y) = b^y$ — base b > 0, b ≠ 1.

Base (b)

Logarithm value (y)

⚠️For educational and informational purposes only. Verify with a qualified professional.

📋 Key Takeaways

• Antilog is the inverse of logarithm: if $\log_b(x) = y$ , then $x = b^y = \text{antilog}_b(y)$
• pH to concentration: $[H^+] = 10^{-\text{pH}}$ — antilog of negative pH
• Decibels: Power ratio = 10^(dB/10). Intensity ratio = 10^(dB/20) for sound
• Richter scale: Amplitude ∝ 10^magnitude. Each whole number = 10× amplitude
• Exponential growth: $A = P \cdot b^t$ — antilog gives the growth factor

💡 Did You Know?

🧪pH 3 means [H⁺] = 10^-3 = 0.001 M. pH 7 is neutral: [H⁺] = 10^-7 M. Antilog converts pH back to concentrationSource: Chemistry

🔊A 20 dB increase means 10^(20/10) = 100× more power. 60 dB is 10^6× the reference intensitySource: Acoustics

🌍Richter 7 is 10× the amplitude of Richter 6. Energy release scales as ~10^(1.5×magnitude)Source: Seismology

📈Log returns in finance: if ln(P₂/P₁)=r, then P₂/P₁=e^r. Antilog gives the price ratioSource: Finance

⚛️Half-life: N/N₀ = 0.5 = e^(-λt½). So e^(-0.693) ≈ 0.5. ln(2) ≈ 0.693 is the decay constant factorSource: Physics

📊Order of magnitude: 10^7 = 10 million, 10^-2 = 0.01. Antilog shows the actual scaleSource: Science

🔬Benford's Law: first digit d appears with probability log₁₀(1+1/d). Antilog relates to digit distributionSource: Statistics

📖 How Antilogarithms Work

Antilog reverses the logarithm. If $\log_b(x) = y$ , then $x = b^y$ . Computing antilog is just exponentiation.

Inverse Relationship

log and antilog are inverse functions: $b^{\log_b(x)} = x$ and $\log_b(b^y) = y$ . Antilog_b(y) = b^y.

pH and Chemistry

pH = -log₁₀([H⁺]), so [H⁺] = 10^(-pH). To find concentration from pH, compute 10^(-pH) — that's antilog base 10 of -pH.

Decibel Conversion

dB = 10·log₁₀(P/P₀) for power. So P/P₀ = 10^(dB/10). For sound intensity: I/I₀ = 10^(dB/10). Antilog converts dB to ratio.

🎯 Expert Tips

💡 pH to [H⁺]

pH 7 → [H⁺]=10^-7 M. pH 3 → 10^-3=0.001 M (1000× more acidic). Each pH unit = 10× concentration change.

💡 Handling Large/Small Results

10^10 = 10 billion. 10^-5 = 0.00001. Use scientific notation. Order of magnitude = floor(log₁₀|x|).

💡 Negative Exponents

10^-2 = 1/100 = 0.01. 2^-3 = 1/8. Negative exponent = fraction. Common in decay and pH.

💡 Base e (Natural Antilog)

antilog_e(y) = e^y = exp(y). Use for continuous growth, half-life (e^(-0.693)≈0.5), and log returns.

⚖️ Why Use This Calculator?

Feature	This Calculator	TI-84 / Casio	Wolfram Alpha
Any base antilog b^y	✅	⚠️ 10^x only	✅
Scientific notation	✅	✅	✅
Order of magnitude	✅	❌	⚠️ Paid
Exponential growth chart	✅	❌	✅
Base comparison chart	✅	❌	❌
Copy & share results	✅	❌	⚠️ Limited
pH/decibel content	✅	❌	⚠️ Limited
Free, no login	✅	N/A	⚠️ Limited

❓ Frequently Asked Questions

Is antilog the same as exponent?

Yes. antilog_b(y) = b^y. "Antilog" emphasizes that you are reversing a logarithm: if log_b(x)=y, then x=b^y. It's exponentiation with the base and exponent swapped in perspective.

How do I find [H⁺] from pH?

[H⁺] = 10^(-pH). For pH 5, [H⁺] = 10^-5 = 0.00001 M. For pH 2.3, [H⁺] = 10^-2.3 ≈ 0.005 M. Use antilog base 10 of -pH.

What is the antilog of a negative number?

b^(-y) = 1/(b^y). So 10^(-2) = 0.01, e^(-1) ≈ 0.368. Negative antilog gives a fraction between 0 and 1.

How do I convert decibels to a power ratio?

Power ratio = 10^(dB/10). So 20 dB → 10^2 = 100× power. For sound intensity (which uses same formula): 60 dB → 10^6× reference.

Why can't the base be 1?

For antilog (b^y), base 1 works (1^y=1 always) but we exclude it to match logarithm domain where log base 1 is undefined. For consistency with log inverse, we require b>0, b≠1.

When to use base 10 vs base e?

Use base 10 for pH, decibels, Richter, scientific notation. Use base e (exp) for continuous growth, half-life, log returns, and calculus.

📊 Key Antilog Values

10^3 = 1000

Common antilog

e^1 ≈ 2.718

Natural antilog

10^-7 M

Neutral pH [H⁺]

10^(dB/10)

Decibel to ratio

📚 Reference Sources

Khan Academy — Exponentials ↗

Interactive lessons on exponentials

Wolfram MathWorld — Antilogarithm ↗

Mathematical reference

Wikipedia — Antilogarithm ↗

Logarithm identities and antilog

NIST — Exponential ↗

DLMF Exponential function

⚠️ Note: This calculator uses IEEE 754 double-precision. Very large exponents may overflow to Infinity. Very small results may underflow. Results are accurate to ~15–17 significant digits.

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