Question

2.13a exponential and logarithmic equations and inequalities
ap precalculus
2.13a practice
calculator active. solve each equation. have both the exact answer and the answer rounded to three decimal places.

1. $\log_{5} x = 2$
2. $e^{-x} = 3.65$
3. $\log x + \log(x - 3) = 1$
4. $80e^{0.045x} = 240$

Explanation:

Step1: Convert log to exponential form

$x = 5^2$

Step2: Calculate exact value

$x = 25$

Step3: Rounded value (no decimals needed)

$x = 25.000$

Step1: Take natural log of both sides

$-x = \ln(3.65)$

Step2: Solve for x (exact)

$x = -\ln(3.65)$

Step3: Calculate rounded value

$x \approx -1.294$

Step1: Combine logs using product rule

$\log[x(x-3)] = 1$

Step2: Convert to exponential (base 10)

$x(x-3) = 10^1 = 10$

Step3: Form quadratic equation

$x^2 - 3x - 10 = 0$

Step4: Factor quadratic

$(x-5)(x+2) = 0$ → $x=5$ or $x=-2$ (discard $x=-2$, log domain)

Step5: Rounded value

$x = 5.000$

Step1: Divide both sides by 80

$e^{0.045x} = 3$

Step2: Take natural log of both sides

$0.045x = \ln(3)$

Step3: Solve for x (exact)

$x = \frac{\ln(3)}{0.045}$

Step4: Calculate rounded value

$x \approx 24.414$

Answer:

Exact: 25, Rounded: 25.000