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 base of the common logarithm is 10.

If $$log_b(M) = log_b(N)$$ , then $$M = N$$ .

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

Exponential functions and logarithmic functions are inverses of each other. If $$y = b^x$$ , then $$x = log_b(y)$$ .

First, isolate the exponential: $$2^{(x-5)} = 24$$ . Then take the logarithm: $$x - 5 = log_2(24)$$ , leading to $$x = log_2(24) + 5$$ .

The product rule states that $$log_b(MN) = log_b(M) + log_b(N)$$ .

The quotient rule states that $$log_b(M/N) = log_b(M) - log_b(N)$$ .

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

Set the arguments equal: $$3x - 1 = 2x + 3$$ . Solve for $$x$$ to find $$x = 4$$ .