AP Precalculus: Exponential & Logarithmic Equations Formulas
1. Solve Exponential Equations by Rewriting the Base
- Express both sides with the same base: \( a^{f(x)} = a^{g(x)} \) → set exponents equal: \( f(x) = g(x) \)
- Example: \( 3^{2x} = 9^{x+1} \implies 3^{2x} = (3^2)^{x+1} = 3^{2x+2} \rightarrow 2x = 2x+2 \)
2. Solve Exponential Equations Using Logarithms
- If unable to rewrite bases, take log of both sides: \( a^{f(x)} = b \implies f(x)\log(a) = \log(b) \)
- General: \( ab^{cx} = d \implies b^{cx} = \frac{d}{a} \implies cx = \log_b\left(\frac{d}{a}\right) \)
- Final formula: \( x = \frac{\log_b\left(\frac{d}{a}\right)}{c} \)
3. Solve Logarithmic Equations with One Log
- \( \log_a(f(x)) = b \implies f(x) = a^b \)
- Solve for \( x \) in result: e.g. \( \log_3(2x-1) = 4 \implies 2x-1=81 \implies x=41 \)
- Check solutions do not make the log undefined (\( f(x)>0 \))
4. Solve Logarithmic Equations with Multiple Logs
- Combine using properties:
- \( \log_a f(x) + \log_a g(x) = \log_a[f(x)g(x)] \)
- \( \log_a f(x) - \log_a g(x) = \log_a\left[\frac{f(x)}{g(x)}\right] \)
- Set equal logs: If \( \log_a f(x) = \log_a g(x) \implies f(x) = g(x) \)
- Check solutions: arguments must be > 0
5. Exponential Growth & Decay
- Growth: \( y = a(1 + r)^t \), \( r > 0 \); Decay: \( y = a(1 - r)^t \), \( r > 0 \)
- Continuous: \( y = ae^{kt} \), growth if \( k > 0 \), decay if \( k < 0 \)
6. Compound Interest
- General formula: \( A = P\left(1+\frac{r}{n}\right)^{nt} \)
- P = initial amount, r = annual interest rate, n = # of compounding periods/year, t = years, A = amount
- Continuous compounding: \( A = Pe^{rt} \)