Exponential, Log and Trig Equations

The basic equations solver handles exponential equations aˣ = b, logarithmic equations logₐx = b and trigonometric equations sin x = a, cos x = a, tan x = a. For standard values (½, √2/2, √3/2, √3 …) the answer is exact in terms of π; for other values it uses arcsin, arccos and arctan with an approximate value.

Exponential equation aˣ = b

• If b ≤ 0, there are no solutions: aˣ > 0 for every x.
• If a and b are powers of the same number, set the exponents equal: 8ˣ = 4 ⇒ 2^3x = 2² ⇒ x = ²/₃.
• Otherwise x = log_a b: 3ˣ = 10 ⇒ x = log₃ 10 ≈ 2.0959.

Logarithmic equation logₐx = b

By the definition of a logarithm, x = aᵇ: log₃ x = 2 ⇒ x = 3² = 9. The base a must be positive and not equal to 1, and x must be positive.

Trigonometric equations

Here n is any integer (n ∈ ℤ). Special cases have shorter forms: sin x = 0 ⇒ x = πn; sin x = 1 ⇒ x = π/2 + 2πn; cos x = 0 ⇒ x = π/2 + πn; cos x = 1 ⇒ x = 2πn.

Standard values

arctan 1 = π/4, arctan √3 = π/3, arctan (√3/3) = π/6. For negative a: arcsin(−a) = −arcsin a, arctan(−a) = −arctan a, arccos(−a) = π − arccos a.