Solving Exponential Equations Using Logarithms

An exponential equation is an equation in which a variable appears in the exponent. For example, $$2^x = 8$$ .

To solve an exponential equation using logarithms, take the logarithm of both sides of the equation. For example, for $$10^x = 200$$ , take $$\text{log}(10^x) = \text{log}(200)$$ .

A logarithm is the inverse operation to exponentiation, indicating the power to which a base must be raised to obtain a given number. For example, $$\text{log}_b(a) = c$$ means $$b^c = a$$ .

The change of base formula states that $$\text{log}_b(a) = \frac{\text{log}_k(a)}{\text{log}_k(b)}$$ for any positive base $$k$$ .

The natural logarithm, denoted as $$\text{ln}(x)$$ , is the logarithm to the base $$e$$ , where $$e$$ is approximately equal to 2.718.

To isolate the variable, use logarithms to rewrite the equation in a form where the variable is no longer in the exponent. For example, from $$2^x = 8$$ to $$x = \text{log}_2(8)$$ .

The formula for continuous compound interest is $$A = Pe^{rt}$$ , where $$A$$ is the amount of money accumulated after time $$t$$ , $$P$$ is the principal amount, $$r$$ is the rate of interest, and $$t$$ is the time.

To find the time it takes for an investment to triple, set $$A = 3P$$ in the formula $$A = Pe^{rt}$$ and solve for $$t$$ .

The value of $$e$$ is approximately 2.71828, and it is the base of the natural logarithm.

1. Take the logarithm of both sides: $$\text{log}(10^x) = \text{log}(200)$$ . 2. Use the property of logarithms: $$x \text{log}(10) = \text{log}(200)$$ . 3. Solve for $$x$$ : $$x = \frac{\text{log}(200)}{\text{log}(10)} \text{ (approximately 2.30)}$$ .