6.6 Exponential and Logarithmic Equations

Exponential Equations

Solving Exponential Equations with Like Bases

When both sides of an equation have the same base, you can drop the bases and set the exponents equal. This works because exponential functions are one-to-one: if $b^m = b^n$, then $m = n$.

1. Rewrite each side so the bases match.
2. Set the exponents equal.
3. Solve the resulting equation.

Example: Solve $5^x = 5^{2x - 1}$

Since the bases are both 5, set the exponents equal: $x = 2x - 1$

Subtract $x$ from both sides: $0 = x - 1$, so $x = 1$.

Sometimes the bases don't look the same at first, but you can rewrite them. For instance, $8^x = 32$ can be rewritten as $(2^3)^x = 2^5$, which gives $2^{3x} = 2^5$, so $3x = 5$ and $x = \frac{5}{3}$.

Using Logarithms to Solve Exponential Equations

When you can't rewrite both sides with the same base, logarithms are the way forward. Logarithms "undo" exponentials: $\log_b(b^x) = x$.

1. Isolate the exponential expression on one side.
2. Take the logarithm of both sides (natural log or common log both work).
3. Use the power rule to bring the variable out of the exponent.
4. Solve for the variable.

Example: Solve $3^x = 20$

Take the natural log of both sides: $\ln(3^x) = \ln(20)$

Apply the power rule: $x \cdot \ln(3) = \ln(20)$

Divide: $x = \frac{\ln(20)}{\ln(3)} \approx 2.727$

Definition of Logarithms in Equations

The key relationship to internalize: $\log_b(x) = y$ means exactly the same thing as $b^y = x$, where $b > 0$, $b \neq 1$, and $x > 0$.

Converting between these two forms is often the entire strategy for solving a logarithmic equation.

Example: Solve $\log_3(x) = 4$

Rewrite in exponential form: $3^4 = x$, so $x = 81$.

The common logarithm ($\log$) has base 10, and the natural logarithm ($\ln$) has base $e \approx 2.718$. These are the two you'll use most often.

One-to-One Property for Logarithmic Equations

Just as equal bases let you set exponents equal, equal logarithms (with the same base) let you set the arguments equal:

If $\log_b(M) = \log_b(N)$, then $M = N$ (provided $b > 0$, $b \neq 1$, and both $M, N > 0$).

1. Use log properties to condense each side into a single logarithm.
2. If both sides have the same base log, set the arguments equal.
3. Solve the resulting equation.
4. Check your solutions in the original equation. You must reject any value that makes an argument of a logarithm zero or negative.

Example: Solve $\log(3x + 1) = \log(x + 5)$

Set the arguments equal: $3x + 1 = x + 5$

Solve: $2x = 4$, so $x = 2$.

Check: $\log(3(2) + 1) = \log(7)$ and $\log(2 + 5) = \log(7)$. Both arguments are positive, so $x = 2$ is valid.

Properties of Logarithms

These three rules let you break apart or combine logarithmic expressions, which is essential for solving equations:

• Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$
• Quotient Rule: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$
• Power Rule: $\log_b(x^n) = n \cdot \log_b(x)$

The power rule is especially useful for exponential equations because it pulls the variable down from the exponent. The product and quotient rules help you condense multiple log terms into one, which sets you up to use the one-to-one property or convert to exponential form.

Real-World Applications of Exponential Equations

Exponential growth and decay follow the general model:

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

where $A_0$ is the initial amount, $k$ is the rate (positive for growth, negative for decay), and $t$ is time. Solving for $t$ in these problems almost always requires taking a logarithm.

For example, if a bacteria colony starts at 500 and doubles every 3 hours, you could find the population at any time, or find when it reaches a target by solving for $t$.

Half-life is the time for a quantity to decrease by half. If a medication has a half-life of 4 hours and you start with 200 mg, after 4 hours you have 100 mg, after 8 hours you have 50 mg, and so on. Doubling time is the growth equivalent: the time for a quantity to double.

Logarithmic scales compress huge ranges of values into manageable numbers:

• The Richter scale measures earthquake magnitude (each whole number is 10× more ground motion)
• The pH scale measures acidity (each unit is a 10× change in hydrogen ion concentration)
• The decibel scale measures sound intensity

Inverse Functions

Exponential and logarithmic functions are inverses of each other. This means:

• $b^{\log_b(x)} = x$ for $x > 0$
• $\log_b(b^x) = x$ for all real $x$

This inverse relationship is why logarithms can solve exponential equations and vice versa. Whenever you're stuck on one form, try converting to the other.