Solving Exponential and Logarithmic Equations

Solving Exponential Equations

An exponential equation has the variable in the exponent.

Strategy 1: Same Base Method

If you can express both sides with the same base, set the exponents equal.

If $b^x = b^y$, then $x = y$

Example

$2^{3x} = 2^{15}$ $3x = 15$ $x = 5$

Strategy 2: Taking Logarithms

If you can't easily get the same base, take the logarithm of both sides.

Steps:

1. Isolate the exponential expression
2. Take log (or ln) of both sides
3. Use the power rule: $\log(b^x) = x\log(b)$
4. Solve for the variable

Example

$5^x = 17$ $\ln(5^x) = \ln(17)$ $x\ln(5) = \ln(17)$ $x = \frac{\ln(17)}{\ln(5)}$

Solving Logarithmic Equations

A logarithmic equation contains logarithmic expressions.

Strategy 1: Convert to Exponential Form

Use the definition: $\log_b(x) = y$ means $b^y = x$

Example

$\log_3(x) = 4$ $x = 3^4 = 81$

Strategy 2: Combine Logarithms

Use logarithm properties to combine into a single log, then solve.

Key Properties to Use

• $\log_b(M) + \log_b(N) = \log_b(MN)$
• $\log_b(M) - \log_b(N) = \log_b(M/N)$
• $p\log_b(M) = \log_b(M^p)$

Strategy 3: Equal Logs Method

If $\log_b(M) = \log_b(N)$, then $M = N$

(assuming same base and same domain)

Important Reminders

⚠️ Check your answers!

• Logarithms require positive arguments: $x > 0$ for $\log(x)$
• Reject any solutions that give $\log(\text{negative})$ or $\log(0)$

⚠️ One-to-one property

• $b^x$ is one-to-one (equal outputs → equal inputs)
• $\log_b(x)$ is one-to-one (equal outputs → equal inputs)

Common Equations to Recognize

Type 1: $b^x = a$

$x = \log_b(a) = \frac{\ln(a)}{\ln(b)}$

Type 2: $a \cdot b^{cx} + d = e$

1. Isolate: $b^{cx} = \frac{e - d}{a}$
2. Take log: $cx\ln(b) = \ln\left(\frac{e - d}{a}\right)$
3. Solve: $x = \frac{1}{c\ln(b)}\ln\left(\frac{e - d}{a}\right)$

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

1. Combine: $\log(x(x - 3)) = 1$
2. Convert: $x(x - 3) = 10^1$
3. Solve quadratic: $x^2 - 3x - 10 = 0$

Applications

1. Compound Interest: $A = P(1 + r)^t$
2. Exponential Growth/Decay: $A = A_0e^{kt}$
3. Half-life Problems: $A = A_0(1/2)^{t/h}$
4. Doubling Time: Solve $2A_0 = A_0e^{kt}$ for $t$