diff --git a/src/function/arithmetic/log.js b/src/function/arithmetic/log.js
index 3dadeaa40e..262f1f225c 100644
--- a/src/function/arithmetic/log.js
+++ b/src/function/arithmetic/log.js
@@ -3,10 +3,10 @@ import { promoteLogarithm } from '../../utils/bigint.js'
 import { logNumber } from '../../plain/number/index.js'
 
 const name = 'log'
-const dependencies = ['config', 'typed', 'typeOf', 'divideScalar', 'Complex']
+const dependencies = ['config', 'typed', 'typeOf', 'divideScalar', 'Complex', 'BigNumber']
 const nlg16 = Math.log(16)
 
-export const createLog = /* #__PURE__ */ factory(name, dependencies, ({ typed, typeOf, config, divideScalar, Complex }) => {
+export const createLog = /* #__PURE__ */ factory(name, dependencies, ({ typed, typeOf, config, divideScalar, Complex, BigNumber }) => {
   /**
    * Calculate the logarithm of a value.
    *
@@ -60,7 +60,17 @@ export const createLog = /* #__PURE__ */ factory(name, dependencies, ({ typed, t
       }
     },
 
-    bigint: promoteLogarithm(nlg16, logNumber, config, complexLogNumber),
+    bigint: function (b) {
+      if (config.number === 'bigint' && config.numberFallback === 'BigNumber') {
+        const x = new BigNumber(b.toString())
+        if (!x.isNegative() || config.predictable) {
+          return x.ln()
+        } else {
+          return complexLogNumber(x.toNumber())
+        }
+      }
+      return promoteLogarithm(nlg16, logNumber, config, complexLogNumber)(b)
+    },
 
     Complex: complexLog,
 
diff --git a/src/function/arithmetic/log10.js b/src/function/arithmetic/log10.js
index 8cde9a3d27..c006b7b36d 100644
--- a/src/function/arithmetic/log10.js
+++ b/src/function/arithmetic/log10.js
@@ -4,10 +4,10 @@ import { deepMap } from '../../utils/collection.js'
 import { factory } from '../../utils/factory.js'
 
 const name = 'log10'
-const dependencies = ['typed', 'config', 'Complex']
+const dependencies = ['typed', 'config', 'Complex', 'BigNumber']
 const log16 = log10Number(16)
 
-export const createLog10 = /* #__PURE__ */ factory(name, dependencies, ({ typed, config, Complex }) => {
+export const createLog10 = /* #__PURE__ */ factory(name, dependencies, ({ typed, config, Complex, BigNumber }) => {
   /**
    * Calculate the 10-base logarithm of a value. This is the same as calculating `log(x, 10)`.
    *
@@ -51,7 +51,17 @@ export const createLog10 = /* #__PURE__ */ factory(name, dependencies, ({ typed,
       }
     },
 
-    bigint: promoteLogarithm(log16, log10Number, config, complexLogNumber),
+    bigint: function (b) {
+      if (config.number === 'bigint' && config.numberFallback === 'BigNumber') {
+        const x = new BigNumber(b.toString())
+        if (!x.isNegative() || config.predictable) {
+          return x.log()
+        } else {
+          return complexLogNumber(x.toNumber())
+        }
+      }
+      return promoteLogarithm(log16, log10Number, config, complexLogNumber)(b)
+    },
 
     Complex: complexLog,
 
diff --git a/src/function/arithmetic/log2.js b/src/function/arithmetic/log2.js
index 19b239f7cf..3eab422f25 100644
--- a/src/function/arithmetic/log2.js
+++ b/src/function/arithmetic/log2.js
@@ -4,9 +4,9 @@ import { deepMap } from '../../utils/collection.js'
 import { factory } from '../../utils/factory.js'
 
 const name = 'log2'
-const dependencies = ['typed', 'config', 'Complex']
+const dependencies = ['typed', 'config', 'Complex', 'BigNumber']
 
-export const createLog2 = /* #__PURE__ */ factory(name, dependencies, ({ typed, config, Complex }) => {
+export const createLog2 = /* #__PURE__ */ factory(name, dependencies, ({ typed, config, Complex, BigNumber }) => {
   /**
    * Calculate the 2-base of a value. This is the same as calculating `log(x, 2)`.
    *
@@ -46,7 +46,17 @@ export const createLog2 = /* #__PURE__ */ factory(name, dependencies, ({ typed,
       }
     },
 
-    bigint: promoteLogarithm(4, log2Number, config, complexLog2Number),
+    bigint: function (b) {
+      if (config.number === 'bigint' && config.numberFallback === 'BigNumber') {
+        const x = new BigNumber(b.toString())
+        if (!x.isNegative() || config.predictable) {
+          return x.log(2)
+        } else {
+          return complexLog2Number(x.toNumber())
+        }
+      }
+      return promoteLogarithm(4, log2Number, config, complexLog2Number)(b)
+    },
 
     Complex: _log2Complex,
 
diff --git a/src/utils/bigint.js b/src/utils/bigint.js
index 4a163cfdfa..6763733cc7 100644
--- a/src/utils/bigint.js
+++ b/src/utils/bigint.js
@@ -16,12 +16,15 @@
  */
 export function promoteLogarithm (log16, numberLog, config, cplx) {
   return function (b) {
+    if (b === 0n) {
+      return numberLog(0)
+    }
     if (b > 0 || config.predictable) {
-      if (b <= 0) return NaN
+      if (b < 0) return NaN
       const s = b.toString(16)
       const s15 = s.substring(0, 15)
       return log16 * (s.length - s15.length) + numberLog(Number('0x' + s15))
     }
-    return cplx(b.toNumber())
+    return cplx(Number(b))
   }
 }
diff --git a/test/unit-tests/function/arithmetic/log-bigint-compat.test.js b/test/unit-tests/function/arithmetic/log-bigint-compat.test.js
new file mode 100644
index 0000000000..e42ec30ebf
--- /dev/null
+++ b/test/unit-tests/function/arithmetic/log-bigint-compat.test.js
@@ -0,0 +1,70 @@
+// Regression tests for bigint + high-precision floating point arithmetic (#3539)
+import assert from 'assert'
+
+import { approxDeepEqual } from '../../../../tools/approx.js'
+import math from '../../../../src/defaultInstance.js'
+
+describe('log-bigint-compat', function () {
+  it('log10(0n) returns -Infinity (no crash)', function () {
+    assert.strictEqual(math.log10(0n), -Infinity)
+  })
+
+  describe('with number: bigint and numberFallback: BigNumber', function () {
+    const m = math.create({
+      number: 'bigint',
+      numberFallback: 'BigNumber',
+      precision: 80
+    })
+
+    it('log returns BigNumber for positive bigint', function () {
+      const r = m.log(123n)
+      assert.strictEqual(m.typeOf(r), 'BigNumber')
+      // sanity: compare to double for closeness (not equality)
+      approxDeepEqual(Number(r.toString()), Math.log(123))
+    })
+
+    it('log10(10n) -> BigNumber(1)', function () {
+      const r = m.log10(10n)
+      assert.strictEqual(m.typeOf(r), 'BigNumber')
+      assert.deepStrictEqual(r, m.bignumber(1))
+    })
+
+    it('log2(1024n) -> BigNumber(10)', function () {
+      const r = m.log2(1024n)
+      assert.strictEqual(m.typeOf(r), 'BigNumber')
+      assert.deepStrictEqual(r, m.bignumber(10))
+    })
+
+    it('floor(log10(1n)) -> BigNumber(0) without round error', function () {
+      const r = m.floor(m.log10(1n))
+      assert.strictEqual(m.typeOf(r), 'BigNumber')
+      assert.deepStrictEqual(r, m.bignumber(0))
+    })
+
+    it('negative bigint: predictable true -> NaN', function () {
+      const mp = m.create({ predictable: true })
+      const r = mp.log(-5n)
+      assert.strictEqual(typeof r, 'number')
+      assert.ok(Number.isNaN(r))
+    })
+
+    it('negative bigint: predictable false -> Complex (downgraded)', function () {
+      const r = m.log(-5n)
+      assert.strictEqual(m.typeOf(r), 'Complex')
+    })
+  })
+
+  describe('with number: bigint and numberFallback: number', function () {
+    const m = math.create({
+      number: 'bigint',
+      numberFallback: 'number'
+    })
+
+    it('promoteLogarithm path returns JS numbers', function () {
+      // log2 of a big power should be exact
+      assert.strictEqual(m.log2(2n ** 70n), 70)
+      // log10 of 10^16
+      assert.strictEqual(m.log10(10n ** 16n), 16)
+    })
+  })
+})