Solving Exponential and Logarithmic Equations

When the variable you are solving for is trapped in an exponent, such as in $5^x = 42$, simple arithmetic won't help. We need a tool to 'pull down' that exponent. This is where the Power Property of Logarithms comes in: $\log_b(m^n) = n \log_b(m)$. By taking the logarithm of both sides of an equation, we transform an exponential problem into a linear one. Whether you use the common log ($\\log_{10}$) or the natural log ($\\ln$), the result remains the same due to the Change of Base Formula. This technique is the foundation for calculating interest rates, population growth, and even the intensity of earthquakes.

Solving logarithmic equations often involves 'exponentiating' both sides to cancel out the log. However, there is a catch: the domain of a logarithmic function $y = \log_b(x)$ is strictly $x > 0$. You cannot take the log of a negative number or zero. When you use properties like the Product Property $\log(a) + \log(b) = \log(ab)$, you might inadvertently create a quadratic equation that yields two solutions. You must check if these solutions, when plugged back into the original equation, result in taking the log of a non-positive number. If they do, they are extraneous solutions and must be discarded.

Logarithms are essential for calculating half-life, the time required for a quantity to reduce to half its initial value. The general formula is $N(t) = N_0(0.5)^{\frac{t}{h}}$, where $N_0$ is the initial amount, $h$ is the half-life, and $t$ is time. To find how long it takes for a substance to decay to a specific percentage, we must solve for $t$ using logs. This is how archaeologists determine the age of ancient artifacts through Carbon-14 dating. By measuring the remaining isotope, they work backward through the exponential decay curve to find the moment the organism stopped breathing.