Solving Exponential and 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 property states that $$log_b(mn) = log_b(m) + log_b(n)$$ .

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

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

Combine the logs: $$log_2(x(x + 5)) = log_2(3x + 8)$$ , then set the arguments equal: $$x(x + 5) = 3x + 8$$ . Solve for $$x$$ .

Rewrite $$49$$ as $$7^2$$ : $$7^{(3x+4)} = (7^2)^{(2x+1)}$$ . This simplifies to $$7^{(3x+4)} = 7^{(4x+2)}$$ . Set the exponents equal: $$3x + 4 = 4x + 2$$ , leading to $$x = 2$$ .

Rewrite $$81$$ as $$3^4$$ : $$3^{(2x - 6)} = 3^4$$ . Set the exponents equal: $$2x - 6 = 4$$ , leading to $$x = 5$$ .

Rewrite in exponential form: $$x + 3 = 2^4$$ , which simplifies to $$x + 3 = 16$$ . Thus, $$x = 13$$ .