Log Calculator: ln, log10, and Change of Base

This page defines logs, compares ln and log10, derives change of base, lists log laws, connects exponents and scientific notation, and routes homework to DevOkk tools. The Log Calculator evaluates ln, log10, and arbitrary base via change of base. Scientific Calculator has the same keys with manual order. Exponent Calculator handles b^y. Scientific Notation Calculator helps when logs appear inside magnitude and significant-figure problems. No account.

For decay formulas that use ln, see Half-life decay: remaining amount formula. For matrix and quadratic lanes in the same course, see Matrix calculator 2×2 3×3.

Definition: log answers the exponent

For b > 0, b ≠ 1, and x > 0:

log_b(x) = y means b^y = x.

Examples:

• log₂(8) = 3 because 2³ = 8.
• log₁₀(1000) = 3 because 10³ = 1000.
• ln(e⁵) = 5 because e⁵ = e⁵ (ln is log base e).

The base b is the repeated multiplier. The argument x must be positive in real-valued intro courses.

ln versus log10: two calculator keys, two bases

Name	Base	Typical notation	Calculator key
Natural log	e ≈ 2.71828	ln(x), log_e(x)	ln
Common log	10	log₁₀(x), sometimes log(x)	log or LOG

ln(10) ≈ 2.303. log₁₀(10) = 1. ln(1) = 0, log₁₀(1) = 0 for any valid base.

Trap: in some math texts log(x) without a subscript means ln (especially in calculus). In algebra and chemistry, log often means base 10. Read the textbook front matter. When unsure, rewrite in log_b(x) with a subscript.

The Log Calculator should label which base each field uses.

Change of base: one formula for every base

Calculators rarely have a log₂ key. Change of base:

log_b(x) = ln(x) / ln(b)

or

log_b(x) = log₁₀(x) / log₁₀(b)

Example: log₂(32).

ln(32)/ln(2) ≈ 3.4657/0.6931 ≈ 5. Check: 2⁵ = 32.

Example: log₃(81) = ln(81)/ln(3) = 4.

Use the same base in numerator and denominator-do not mix ln on top and log10 on bottom unless you convert consistently (they are proportional).

Log laws: compress before you calculate

For valid positive arguments:

Product: log_b(MN) = log_b(M) + log_b(N)

Quotient: log_b(M/N) = log_b(M) − log_b(N)

Power: log_b(M^k) = k · log_b(M)

Change of base (above) is also a law.

These laws explain why ln(e^x) = x and e^(ln x) = x for x > 0.

Example: expand ln(3x²) = ln 3 + 2 ln x (domain x > 0).

Example: solve 10^(2x) = 1000. Take log₁₀: 2x = 3 → x = 3/2.

Solving equations with logs and exponents

Type A - variable in exponent: isolate b^(something), take log both sides.

2^x = 64 → x log 2 = log 64 (any base) → x = log 64 / log 2 = 6.

Type B - variable inside log: combine logs, rewrite in exponential form.

ln(x − 1) = 2 → x − 1 = e² → x = 1 + e².

Type C - logs on both sides: use laws to collapse.

log(x) + log(3) = log(12) → log(3x) = log(12) → 3x = 12 → x = 4 (domain x > 0).

Always check domain: ln(x − 1) requires x > 1; reject extraneous values from algebra.

The Exponent Calculator verifies the exponential side after you solve.

pH, decibels, and base-10 in science

pH = −log₁₀[H⁺] where [H⁺] is molarity.

If [H⁺] = 1×10⁻⁷, pH = −log(10⁻⁷) = 7.

Reverse: [H⁺] = 10^(−pH).

Decibel scales use log₁₀ of power ratios. The constant factors (10 vs 20) depend on whether power or amplitude-follow the formula sheet.

These are log10 problems, not ln, unless the course redefines them.

ln in calculus and continuous growth

Continuous growth: A = A₀ e^(kt). Doubling time and half-life formulas often use ln 2 and ln(1/2).

If k = 0.05 per year, time to double: t = ln(2)/k ≈ 0.693/0.05 ≈ 13.9 years.

Half-life page: Half-life decay.

Derivative of ln x is 1/x (calculus link). At algebra level, know ln as inverse of e^x.

Scientific notation and logs together

Large numbers: 6.02×10²³. log₁₀(6.02×10²³) = log₁₀(6.02) + 23 ≈ 0.779 + 23 ≈ 23.779.

The characteristic (integer part from the exponent) plus mantissa log is how slide rules and mental estimates worked.

Scientific Notation Calculator converts standard form; pair with log for magnitude comparisons in physics.

Homework workflow with DevOkk tools

Step 1 - Identify base in the problem statement (e, 10, 2, or unspecified log).

Step 2 - Domain: argument > 0.

Step 3 - Apply log laws to simplify before numeric evaluation.

Step 4 - Evaluate with ln/log10 and change of base if needed.

Step 5 - Check by exponentiating back.

Open Log Calculator for ln, log10, and custom base. Open Scientific Calculator when the worksheet is one numeric log after setup. Open Exponent Calculator to verify b^y. Open Scientific Notation Calculator when arguments are written as a×10^n.

Worked examples

Evaluate ln(e³ · 2) = ln(e³) + ln(2) = 3 + ln(2) ≈ 3.693.

Solve log₂(x) = 5 → x = 2⁵ = 32.

Change of base: log₅(125) = ln(125)/ln(5) = 3.

pH: [H⁺] = 2×10⁻⁴ → pH = −log(2×10⁻⁴) = −(log 2 + (−4)) ≈ 3.70.

Logarithmic scales and orders of magnitude

When one quantity spans many powers of ten-sound intensity, earthquake energy, hydrogen ion concentration-a log scale compresses the axis.

Each unit on a log10 axis multiplies the underlying value by 10. Going from pH 3 to pH 4 divides [H⁺] by 10, not by 1.

Natural log appears when the continuous model is e-based: continuous compound interest A = Pe^(rt), radioactive decay N = N₀ e^(−λt). Taking ln of both sides linearizes the exponent:

ln(N/N₀) = −λt → solve for t without guessing.

Compare orders: log10(10⁹) − log10(10⁶) = 9 − 6 = 3 decades difference-a multiplicative factor of 10³ = 1000.

Pair with Scientific Notation Calculator when values are given as 3.2×10^−5 and the question asks which is larger after log transform.

Graphical intuition: ln and log10 as inverse slopes

The graph of y = ln(x) passes through (1, 0) and (e, 1). It rises slowly for large x-why ln differentiates long products in calculus-and steeply near zero (vertical asymptote at x = 0).

y = log10(x) passes through (1, 0) and (10, 1). Same shape, different horizontal stretch: log10(x) = ln(x)/ln(10).

When a problem asks "estimate log(500)," bracket: log(100) = 2, log(1000) = 3, so log(500) is between 2 and 3, closer to 2.7. Calculator exact: ≈ 2.699.

Special values and exact logs to memorize

Certain logs appear often enough that memorizing them saves keystrokes:

x	ln(x)	log₁₀(x)
1	0	0
e	1	-
10	ln(10) ≈ 2.303	1
e²	2	-
10ⁿ	n·ln(10)	n

log_b(1) = 0 always. log_b(b) = 1 always. log_b(b^k) = k.

Example chain: log₂(1/8) = log₂(2^(−3)) = −3 without a calculator.

When solving e^(2x) = 5, take ln: 2x = ln 5 → x = (ln 5)/2. The Log Calculator confirms; the setup is what earns partial credit.

Calculator keystroke order with Scientific Calculator

On Scientific Calculator, ln(3 + 4) requires parentheses: ln(7), not ln(3) + 4.

For ln(2) + ln(5), apply log laws first: ln(10) ≈ 2.303, faster than two keystrokes and an add.

For change of base log₂(50), enter ln(50) ÷ ln(2). Close denominators: (ln(50))/(ln(2)), not ln(50)/ln(2) mis-parsed as ln(50/ln(2)) on some interfaces.

Write the base and argument on paper before tapping keys-most log errors are domain slips (zero or negative inside) or wrong base selection, not arithmetic.

Common traps

log(a + b) ≠ log a + log b.

Taking log of a negative number.

Mixing ln and log10 without change of base.

Forgetting domain when solving ln(x − 2) = 1.

Calculator mode: degrees vs radians does not affect log, but students confuse keys-ln vs log vs 1/x.

Privacy and what the calculator does not do

Log evaluations run in the browser. No account. The tool returns numeric values; it does not substitute for showing log-law steps when the rubric requires expansion or solving with work shown.

The inverse pair you keep after the unit

log_b(x) = y ↔ b^y = x. ln is base e; log10 is base 10. Change of base converts any b using ln(x)/ln(b). Log laws turn products into sums and powers into factors.

Use Log Calculator for natural log and base-10 evaluation. Use Scientific Calculator for the same arithmetic in long worksheets. Use Exponent Calculator and Scientific Notation Calculator when the problem mixes growth, notation, and magnitude. Pair with Half-life decay for ln in decay. Classify the base, check domain, simplify, evaluate, verify.