Evaluate log_b(a)—the exponent that raises base b to a. Common log, natural log, log₂, and change-of-base for pH, decibels, and solving a^x = b.
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log_b(a) answers: to what power must base b be raised to get a? log₁₀(1000) = 3 because 10³ = 1000. Logarithms compress wide ranges—pH, decibels, and Richter scales all use them.
Logarithm is the inverse of exponentiation. If b^x = a, then x = log_b(a). The base b must be positive and not 1; a must be positive for real logarithms.
**Common bases:** log₁₀ (common log) for scientific notation; log_e ≈ ln (natural log) for calculus and continuous growth; log₂ for information and computing bit counts.
**Product rule:** log(xy) = log(x) + log(y). **Power rule:** log(x^n) = n log(x). These turn multiplication into addition—why slide rules worked.
**Change of base:** log_b(a) = ln(a)/ln(b). Any base converts through natural or common log.
**Applications:** pH = −log₁₀[H⁺]; sound intensity dB = 10 log₁₀(I/I₀); solving 2^x = 50 gives x = log(50)/log(2). Exponent calculator raises bases; logarithm calculator finds the exponent.
Enter value and base (or use ln/log10 presets) to evaluate logarithms and apply change-of-base locally.
Concise answers for common searches — definitions, steps, and comparisons.
log₁₀(100) = 2 because 10² = 100.
ln(1) = 0 because e^0 = 1.
log_b(a) = ln(a) / ln(b). Example: log₂(8) = ln(8)/ln(2) = 3.
H⁺ concentrations span huge ranges. pH = −log₁₀[H⁺] compresses them to manageable numbers.
Computes log_b(a) as ln(a)/ln(b) with validation for positive a and valid base b (b > 0, b ≠ 1).
Formula
log_b(a) = ln(a)/ln(b). log(xy) = log x + log y. log(x^n) = n log x. log_b(a) = 1/log_a(b).Use ln, log10, or log2 shortcuts—or enter custom base like 3.
Value a must be > 0 for a real result.
Result is the exponent that raises b to a.
Verify b^(result) ≈ a to confirm.
Input
log₁₀(1000)Output
310³ = 1000.
Input
ln(e)Output
1e¹ = e by definition.
Input
log₂(1024)Output
102^10 = 1024—ten address bits.
Input
log₃(81)Output
43^4 = 81; or ln(81)/ln(3) = 4.
Input
ln(0.5)Output
≈ −0.693ln(1/2) = −ln(2)—half-life decay sign.
Input
x = log₁₀(500)Output
≈ 2.699Exponent x that produces 500 from base 10.
Common real-world scenarios where this tool saves time.
2^x = 100 → x = log(100)/log(2) ≈ 6.644—how many doublings reach 100.
Hydrogen ion concentration 1×10^(−4) M → pH = −log₁₀(10^(−4)) = 4.
Intensity 100× reference: 10 log₁₀(100) = 20 dB increase.
Convert axis values between linear and log10 for science fair data plots.
Step-by-step chains that connect related tools for common tasks.
| x | log₁₀(x) | Note |
|---|---|---|
| 1 | 0 | 10^0 |
| 10 | 1 | One decade |
| 100 | 2 | Two decades |
| 1000 | 3 | Three decades |
| 0.01 | −2 | Reciprocal power |
| Expression | Base | Value (approx) |
|---|---|---|
| ln(e) | e | 1 |
| ln(10) | e | 2.303 |
| log₁₀(2) | 10 | 0.301 |
| log₂(10) | 2 | ≈ 3.322 |
log_b(a) = ln(a)/ln(b) works on any calculator with ln.
log₁₀(45) is between 1 and 2 because 10 < 45 < 100.
Always verify b^x ≈ a after solving for x.
ln(0.01) is negative—concentrations less than 1 M in chemistry.
Real log domain is (0, ∞). log(0) is undefined.
Log converts products to sums, not sums to sums. log(10+10) ≠ log(10)+log(10).
ln is base e; log without subscript in calculus often means ln, in engineering often log10—check context.
b = 1 gives constant 1^x; negative bases break real logs for non-integer exponents.
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Undefined—no real x satisfies e^x = 0.
x = log(200)/log(5) using any valid base in change-of-base formula.
No—inverse operations. log_b(b^a) = a and b^(log_b(a)) = a.
Often log10 for the LOG key and ln for natural log. This tool labels bases explicitly.
Yes—log_b(a) < 0 when 0 < a < 1. Example: ln(0.5) ≈ −0.693.
Log-normal data models multiplicative noise; raw SD on log-transformed data differs from SD on original scale.
Zero—b^0 = 1 for any valid base b.
Log inputs process locally without server upload.
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Reviewed by EverydayTools Editorial Team on 2026-07-03.
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Keys: 1 log₁₀ · L ln · 2 log₂. x must be positive; custom base b ≠ 1.
Base: log₁₀
log_b(x) is the power you raise b to get x. For homework, use quick starts for log₁₀(100), ln(e), and log₂(8)—nothing is uploaded.