Log Equation Calculator

Use the Log Equation Calculator to solve logarithmic equations quickly with accurate step-by-step results.

What is a Log Equation Calculator?

A log equation calculator is a specialized mathematical tool designed to solve equations involving logarithmic functions. Logarithms are the inverse operations of exponentiation, answering the fundamental question: to what exponent must a base be raised to yield a specific number? When dealing with single logarithms, natural logs, or complex algebraic expressions containing nested logarithmic properties, an online log equation solver simplifies steps, finds real roots, and identifies extraneous solutions that fall outside valid domain restrictions.

Logarithmic Equations Formulas

The foundational relationship between a logarithmic statement and its exponential equivalent is given by:

$$\log_b(x) = y \iff b^y = x \quad \text{where } b > 0, b \neq 1, \text{ and } x > 0$$

Essential Logarithm Rules for Solving Equations

To combine or expand terms when isolating variables, apply these standard logarithm identities:

$$\text{Product Rule: } \log_b(M \cdot N) = \log_b(M) + \log_b(N)$$
$$\text{Quotient Rule: } \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$$
$$\text{Power Rule: } \log_b(M^k) = k \cdot \log_b(M)$$
$$\text{Change of Base Formula: } \log_b(x) = \frac{\ln(x)}{\ln(b)} = \frac{\log_{10}(x)}{\log_{10}(b)}$$

Combining Logarithms Using Product Rule

Solve the following logarithmic equation for \(x\):

$$\log_2(x) + \log_2(x - 2) = 3$$

Step-by-Step Solution

First, apply the product rule to condense the left side into a single logarithm:

$$\log_2(x(x - 2)) = 3 \implies \log_2(x^2 - 2x) = 3$$

Convert the equation from logarithmic form to its exponential equivalent:

$$x^2 - 2x = 2^3 \implies x^2 - 2x = 8$$

Rearrange into a standard quadratic equation and factorize:

$$x^2 - 2x - 8 = 0 \implies (x - 4)(x + 2) = 0$$

This yields two potential solutions: \(x = 4\) and \(x = -2\). Since the argument of a logarithm must be strictly positive (\(x > 0\)), \(x = -2\) is an extraneous solution. Thus, the only valid solution is \(x = 4\).

Solving Natural Logarithms (\(\ln\)) with Exponentials

Solve the natural logarithm equation for \(t\):

$$4\ln(3t + 1) - 5 = 7$$

Step-by-Step Solution

Isolate the natural logarithmic term on one side of the equation:

$$4\ln(3t + 1) = 12 \implies \ln(3t + 1) = 3$$

Exponentiate both sides using Euler's number (\(e\)) as the base:

$$e^{\ln(3t + 1)} = e^3 \implies 3t + 1 = e^3$$

Solve for $t$ algebraically:

$$3t = e^3 - 1 \implies t = \frac{e^3 - 1}{3} \approx \frac{20.0855 - 1}{3} \approx 6.3618$$

The exact answer is \(t = \frac{e^3 - 1}{3}\) (or approximately \(6.3618\)).