Logarithmic Equation Solver. Solve logarithmic equations step by step using this Logarithmic Equation Solver. Enter your logarithm base, the argument value, and the result (log value) to find the unknown — whether you're solving for x in logb(x) = c or finding the base. Get the exact solution, the equivalent exponential form, and a verification check all in one place. Also try the calculate Natural Logarithm (ln) ln(x).
A logarithmic equation is an equation that contains a logarithm of an unknown variable. For example, log₂(x) = 6 asks: 'To what power must 2 be raised to get x?' The answer is x = 2⁶ = 64. Logarithmic equations are the inverse of exponential equations. See also our log₁₀(x) — Log Base 10.
The standard method is to convert the logarithmic form log_b(x) = c into its equivalent exponential form b^c = x. For instance, log₃(x) = 4 becomes 3⁴ = x, so x = 81. Always check that the argument x is positive in your final answer, as logarithms are only defined for positive values.
log (common logarithm) uses base 10, while ln (natural logarithm) uses base e ≈ 2.71828. Both follow the same rules — ln(x) = log_e(x). To use this solver for natural log, enter 2.71828 as your base. For common log (log₁₀), enter 10 as the base.
No — the base must be a positive real number and cannot equal 1. A base of 1 is invalid because 1 raised to any power always equals 1, making the equation unsolvable. Negative bases and zero are also not permitted in standard real-number logarithms.
A solution is only valid if the argument of every logarithm is positive. After solving algebraically, you must substitute each candidate solution back into the original equation and reject any that make the argument zero or negative. These rejected values are called extraneous solutions.
The change of base formula states: log_b(x) = ln(x) / ln(b) = log(x) / log(b). This is useful because most calculators only have buttons for log (base 10) and ln (base e). You can compute any logarithm by dividing the natural log of the argument by the natural log of the base.
A negative log value means the argument is between 0 and 1 (for bases greater than 1). For example, log₂(0.25) = −2 because 2⁻² = 0.25. Negative log values are completely valid — they simply indicate the argument is a fraction less than 1.
The three main log laws are: (1) Product Rule — log_b(MN) = log_b(M) + log_b(N); (2) Quotient Rule — log_b(M/N) = log_b(M) − log_b(N); (3) Power Rule — log_b(M^p) = p · log_b(M). These rules let you combine or expand logarithmic expressions before solving. You might also find our Compound Growth Calculator useful.