Logarithmic Equation Solver

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 log_b(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 Final Value x(t) — Exponential Growth.

Results

Solution

Equivalent Exponential Form

Verification — log_b(x) =

ln(x) — Natural Log of Argument

log₁₀(x) — Common Log of Argument

Results Table

What is a logarithmic equation?

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 Compound Growth Calculator.

How do you solve a logarithmic equation?

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.

What is the difference between log and ln?

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.

Can the base of a logarithm be any number?

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. You might also find our Exponential Function Calculator useful.

What are valid solutions to a logarithmic equation?

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.

How does the change of base formula work?

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.

What does it mean when the log value is negative?

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.

What are the key logarithm laws used to simplify equations?

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.