Solving Logarithmic Equations

A logarithm is the exponent to which a base must be raised to produce a given number. For example, if $$b^y = x$$ , then $$log_b(x) = y$$ .

The change of base formula states that $$log_b(a) = \frac{log_k(a)}{log_k(b)}$$ for any positive base $$k$$ .

To solve $$log_b(x) = y$$ , rewrite it in exponential form: $$x = b^y$$ .

The product rule states that $$log_b(mn) = log_b(m) + log_b(n)$$ .

The quotient rule states that $$log_b(\frac{m}{n}) = log_b(m) - log_b(n)$$ .

The power rule states that $$log_b(m^p) = p \cdot log_b(m)$$ .

Convert to exponential form: $$x + 3 = 2^4$$ , so $$x = 16 - 3 = 13$$ .

Take the logarithm of both sides: $$(x-1) \cdot log(4) = log(50)$$ , then solve for $$x$$ .

Set the arguments equal: $$x^2 - 6 = 2x + 2$$ , solve to find $$x = 4$$ .

Convert to exponential form: $$4x + 4 = 8^2$$ , then solve for $$x$$ .