4.6 Exponential and Logarithmic Equations

Exponential and logarithmic equations show up whenever you need to model quantities that grow or shrink by a constant percentage: population growth, radioactive decay, compound interest, and more. Solving these equations requires you to move fluently between exponential and logarithmic forms, so the core skill here is understanding that logs and exponentials are inverses of each other.

Exponential Equations

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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 $a^m = a^n$, then $m = n$.

• $2^x = 2^5$ gives you $x = 5$ immediately, since the bases match.
• For something like $3^{2x-1} = 3^{4x+7}$, set the exponents equal:

1. Write $2x - 1 = 4x + 7$

2. Subtract $2x$ from both sides: $-1 = 2x + 7$

3. Subtract 7: $-8 = 2x$

4. Divide: $x = -4$

If the bases don't look the same at first, try rewriting them. For example, $4^x = 8$ can be rewritten as $(2^2)^x = 2^3$, which becomes $2^{2x} = 2^3$, so $2x = 3$ and $x = \frac{3}{2}$.

Logarithms for exponential equations

When you can't rewrite both sides with the same base, take a logarithm of both sides.

For $2^x = 50$:

1. Take the natural log (or common log) of both sides: $\ln(2^x) = \ln(50)$
2. Bring the exponent down using the power rule: $x \cdot \ln(2) = \ln(50)$
3. Solve: $x = \frac{\ln(50)}{\ln(2)} \approx 5.644$

This approach works with $\ln$, $\log$, or any base. You're really using the change of base formula:

$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

This formula lets you evaluate any logarithm on a calculator that only has $\ln$ and $\log_{10}$ buttons. For instance, $\log_2(50) = \frac{\log(50)}{\log(2)}$.

Logarithmic Equations

Definition of logarithms in equations

The key definition to internalize: $\log_b(x) = y$ means $b^y = x$. Converting a logarithmic equation into exponential form is often the fastest way to solve it.

• $\log_3(x) = 4$ converts to $3^4 = x$, so $x = 81$.
• $\log_5(2x + 1) = 2$ converts to $5^2 = 2x + 1$, giving $25 = 2x + 1$, so $x = 12$.

One-to-one property for logarithmic equations

Since logarithmic functions are one-to-one, if $\log_b(x) = \log_b(y)$, then $x = y$. You can use this whenever both sides of an equation are logs with the same base.

For $\log(x + 1) = \log(2x - 3)$:

1. Set the arguments equal: $x + 1 = 2x - 3$

2. Solve: $x = 4$

3. Check for extraneous solutions. Plug $x = 4$ back into the original arguments: $4 + 1 = 5 > 0$ and $2(4) - 3 = 5 > 0$. Both are positive, so $x = 4$ is valid.

That check in step 3 matters. You can't take the log of zero or a negative number, so any solution that makes an argument non-positive must be thrown out. This is one of the most common mistakes on tests.

Properties and Characteristics

Exponential and Logarithmic Properties

These properties are your main tools for rewriting and simplifying equations before solving.

Exponential properties:

• Product rule: $a^x \cdot a^y = a^{x+y}$
• Quotient rule: $\frac{a^x}{a^y} = a^{x-y}$
• Power rule: $(a^x)^y = a^{xy}$

Logarithmic properties:

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

The log power rule is especially useful for solving exponential equations, since it lets you pull the variable out of the exponent.

Domain and Range

Exponential and logarithmic functions have complementary domains and ranges, which makes sense because they're inverses.

• Exponential functions ($y = a^x$, where $a > 0, a \neq 1$): domain is all real numbers; range is $(0, \infty)$
• Logarithmic functions ($y = \log_a(x)$): domain is $(0, \infty)$; range is all real numbers

The restricted domain of logarithmic functions is exactly why you need to check for extraneous solutions.

Applications in Science

Half-life is the time it takes for half of a substance to decay. The formula is:

$A(t) = A_0\left(\frac{1}{2}\right)^{t/t_{1/2}}$

where $A_0$ is the initial amount, $t$ is elapsed time, and $t_{1/2}$ is the half-life. For example, if a substance has a half-life of 5 years and you start with 100 grams, after 15 years you'd have $100\left(\frac{1}{2}\right)^{15/5} = 100\left(\frac{1}{2}\right)^3 = 12.5$ grams.

Applications

Real-world applications of exponential equations

The general continuous growth/decay model is:

$A = A_0 e^{kt}$

where $A_0$ is the initial amount, $k$ is the rate (positive for growth, negative for decay), and $t$ is time. A larger $|k|$ means faster change.

pH scale: pH measures how acidic or basic a solution is, defined as:

$pH = -\log_{10}[H^+]$

where $[H^+]$ is the hydrogen ion concentration in moles per liter. A solution with $[H^+] = 10^{-3}$ has $pH = 3$, which is quite acidic. Notice that each 1-unit decrease in pH means a 10-fold increase in hydrogen ion concentration.

Richter scale: Earthquake magnitude is measured as:

$M = \log_{10}\left(\frac{A}{A_0}\right)$

where $A$ is the maximum seismic wave amplitude and $A_0$ is a reference amplitude. Because of the logarithm, each 1-point increase on the Richter scale represents a 10-fold increase in wave amplitude. This is why a magnitude 7 earthquake is not "a little worse" than a magnitude 6; it's 10 times larger in amplitude.