Exponential and Logarithmic Equations

You have learned about exponential functions and logarithmic functions separately. Now comes the moment when these powerful tools combine forces. In this chapter, we tackle equations where the variable is stuck in an exponent or trapped inside a logarithm. These are not just abstract exercises; they are the equations that describe how long it takes for an investment to double, how much of a radioactive substance remains after a given time, and how scientists determine the age of ancient artifacts.

The good news is that you already have all the tools you need. Logarithms undo exponentials. Exponentials undo logarithms. The key to solving these equations is knowing when and how to apply each tool. Once you see the pattern, you will find that even intimidating-looking equations become quite manageable.

Core Concepts

Solving Exponential Equations Using Logarithms

When you encounter an equation like $2^x = 10$, the variable is in the exponent. You cannot use the same-base method from the previous chapter because 10 is not a nice power of 2. This is where logarithms become essential.

The Strategy: Take the logarithm of both sides, then use the power rule to bring the exponent down where you can solve for it.

Here is the general approach:

1. Isolate the exponential expression on one side of the equation.
2. Take the logarithm of both sides (either $\log$ or $\ln$ works).
3. Apply the power rule: $\log(a^x) = x \cdot \log(a)$.
4. Solve for the variable.

Why does this work? Because logarithms and exponentials are inverses. Taking the log of an exponential expression “undoes” the exponential, pulling the exponent down to ground level where you can work with it using basic algebra.

Solving Logarithmic Equations by Converting to Exponential Form

When the variable is inside a logarithm, you need the opposite strategy: convert to exponential form to “free” the variable.

For an equation like $\log_2(x) = 5$, recall the fundamental relationship:

$$\log_b(x) = y \quad \text{means} \quad b^y = x$$

So $\log_2(x) = 5$ becomes $2^5 = x$, giving us $x = 32$.

The Strategy:

1. Isolate the logarithm on one side of the equation.
2. Convert to exponential form using the definition.
3. Solve for the variable.
4. Check your answer in the original equation.

Solving Equations with Logarithms on Both Sides

Sometimes you will encounter equations with logarithms on both sides, like $\log_3(x + 4) = \log_3(2x - 1)$. These use a different property.

The One-to-One Property: If $\log_b(M) = \log_b(N)$, then $M = N$ (provided $M > 0$ and $N > 0$).

This property comes from the fact that logarithmic functions are one-to-one: each output corresponds to exactly one input. If two things have the same logarithm (with the same base), they must be equal.

The Danger of Extraneous Solutions

Here is something critically important: always check your solutions in the original equation.

When solving logarithmic equations, you may find solutions that make mathematical sense algebraically but violate the domain restrictions of logarithms. Remember, you cannot take the logarithm of zero or a negative number. A solution that results in a negative argument is called an extraneous solution and must be rejected.

This is not just a technicality. Extraneous solutions arise naturally when you square both sides of an equation or when you use logarithm properties to combine terms. Always verify that your solutions actually work.

Notation and Terminology

Term	Meaning
Exponential equation	An equation where the variable appears in an exponent, such as $3^x = 81$
Logarithmic equation	An equation where the variable appears inside a logarithm, such as $\log(x) = 2$
Extraneous solution	A solution obtained algebraically that does not satisfy the original equation
Half-life	The time required for a quantity to reduce to half its initial value
Doubling time	The time required for a quantity to double
Decay constant	The rate constant $k$ in exponential decay: $A(t) = A_0 e^{-kt}$
Growth constant	The rate constant $k$ in exponential growth: $A(t) = A_0 e^{kt}$
Exponential regression	Fitting an exponential model to data points

Solving Exponential Equations

Method 1: Same Base (Review)

When you can express both sides as powers of the same base, set the exponents equal.

Method 2: Taking Logarithms

When the same-base method does not work, take the logarithm of both sides.

Solving Logarithmic Equations

Method 1: Convert to Exponential Form

When you have a single logarithm equal to a number, convert to exponential form.

Method 2: Use Logarithm Properties to Combine

When multiple logarithms appear, use properties to combine them into a single logarithm.

Method 3: One-to-One Property

When logarithms with the same base appear on both sides, set the arguments equal.

Key Properties and Rules

Exponential Equation Strategies

Situation	Strategy
Both sides can be written with same base	Set exponents equal
Cannot use same base	Take logarithm of both sides
Equation looks like quadratic in $e^x$ or $b^x$	Substitute $u = e^x$ or $u = b^x$
Multiple exponential terms	Try to factor or combine

Logarithmic Equation Strategies

Situation	Strategy
Single log equals a number	Convert to exponential form
Multiple logs on one side	Combine using product/quotient rules, then convert
Logs on both sides with same base	Use one-to-one property
Logs with different bases	Use change of base formula

Critical Reminder: Checking Solutions

For logarithmic equations, always verify:

1. All arguments of logarithms must be positive.
2. The solution must satisfy the original equation.

For exponential equations, checking is usually straightforward since exponentials are defined for all real exponents.

Applications: Exponential Growth and Decay

The General Model

Many real-world phenomena follow the exponential model:

$$A(t) = A_0 \cdot e^{kt}$$

Where:

• $A(t)$ is the amount at time $t$
• $A_0$ is the initial amount (when $t = 0$)
• $k$ is the rate constant (positive for growth, negative for decay)
• $t$ is time

This formula appears in population growth, radioactive decay, drug metabolism, and continuously compounded interest.

Finding Half-Life

Half-life is the time it takes for a quantity to reduce to half its initial value. It is a constant for any given substance, regardless of the starting amount.

To find half-life, set $A(t) = \frac{1}{2}A_0$ and solve for $t$:

$$\frac{1}{2}A_0 = A_0 \cdot e^{-kt}$$ $$\frac{1}{2} = e^{-kt}$$ $$\ln\left(\frac{1}{2}\right) = -kt$$ $$-\ln(2) = -kt$$ $$t = \frac{\ln(2)}{k}$$

So the half-life formula is:

$$t_{1/2} = \frac{\ln(2)}{k} \approx \frac{0.693}{k}$$

Finding Doubling Time

Doubling time is the time it takes for a growing quantity to double. The derivation is similar to half-life.

Set $A(t) = 2A_0$ and solve:

$$2A_0 = A_0 \cdot e^{kt}$$ $$2 = e^{kt}$$ $$\ln(2) = kt$$ $$t = \frac{\ln(2)}{k}$$

Notice this is the same formula as half-life! The difference is whether $k$ represents growth or decay.

Exponential Regression: Fitting Data

In real-world applications, you often have data points and need to find an exponential model that fits them. This is called exponential regression.

The Basic Idea

Given data that appears to follow exponential growth or decay, you want to find values of $a$ and $b$ (or $a$ and $k$) so that $y = a \cdot b^x$ or $y = a \cdot e^{kx}$ best fits the data.

The Logarithmic Linearization Method

Here is a clever technique: if $y = a \cdot b^x$, then taking the logarithm of both sides gives:

$$\log(y) = \log(a) + x \cdot \log(b)$$

This is a linear equation! If you plot $\log(y)$ versus $x$, you get a straight line with:

• Slope = $\log(b)$
• $y$-intercept = $\log(a)$

This means you can use linear regression techniques on the transformed data to find exponential models.

Using Two Points

If you have two data points, you can find the exponential model exactly.

Real-World Applications

Radioactive Decay in Medicine

Radioactive isotopes are used in medical imaging and treatment. Understanding their decay is crucial for proper dosing.

Iodine-131, used to treat thyroid conditions, has a half-life of about 8 days. If a patient receives a 100-millicurie dose, the amount remaining after $t$ days is:

$$A(t) = 100 \cdot \left(\frac{1}{2}\right)^{t/8} = 100 \cdot e^{-0.0866t}$$

After 24 days (three half-lives), only $100 \times (1/2)^3 = 12.5$ millicuries remain.

Population Dynamics

World population growth can be modeled exponentially (at least over certain periods). If the growth rate is 1.1% per year:

$$P(t) = P_0 e^{0.011t}$$

Doubling time: $\frac{\ln(2)}{0.011} \approx 63$ years.

Drug Metabolism

When you take medication, your body eliminates it exponentially. If a drug has a half-life of 4 hours:

$$C(t) = C_0 \cdot \left(\frac{1}{2}\right)^{t/4}$$

This explains why some medications must be taken every 4-6 hours to maintain therapeutic levels.

Newton’s Law of Cooling

A hot object cools according to:

$$T(t) = T_{\text{room}} + (T_0 - T_{\text{room}}) \cdot e^{-kt}$$

Forensic scientists use this to estimate time of death by measuring body temperature.

Financial Planning

The formula $A = Pe^{rt}$ (continuous compounding) helps answer questions like:

• How long to reach a savings goal?
• What interest rate is needed to double money in 10 years?
• How much to invest now to have a certain amount later?

Self-Test Problems

Summary

• Exponential equations have the variable in the exponent. Solve them by using the same-base method when possible, or by taking logarithms of both sides.

• Logarithmic equations have the variable inside a logarithm. Solve them by converting to exponential form, combining logarithms using properties, or applying the one-to-one property.

• Extraneous solutions can arise when solving logarithmic equations. Always check that your solutions produce positive arguments for all logarithms in the original equation.

• The power rule for logarithms ($\log(a^x) = x \cdot \log(a)$) is the key tool for solving exponential equations, as it brings the variable out of the exponent.

• Half-life and doubling time both use the formula $t = \frac{\ln(2)}{k}$, where $k$ is the decay or growth constant.

• The exponential model $A(t) = A_0 e^{kt}$ describes many real-world phenomena:

  • Growth when $k > 0$ (populations, investments)
  • Decay when $k < 0$ (radioactive decay, drug metabolism)

• Exponential regression fits exponential models to data. With two points, you can find exact values for the parameters. With more points, you can use linearization or technology.

• These equations appear everywhere in science and finance: calculating the age of artifacts, determining drug dosages, projecting population growth, and planning investments.

• The interplay between exponentials and logarithms is one of the most powerful tools in mathematics. Mastering these equation-solving techniques prepares you for calculus, where these functions take center stage.