Growth and Decay

Solving Exponential Equations

200

Solve for x. Express your solution in exact form.

7^{2x}=\frac{1}{49}

x=-1

300

Solve for x. Express your answer in exact form,

2\cdot 3^x = 8\cdot (\frac{1}{9})^{x+1}

x = -\frac{1}{3}+\frac{1}{3}\log_3 4

400

Solve for x. Express your answer in exact form.

2e^x + 3e^{-x} = 7

x = \ln\frac{1}{2}

x = \ln 3

500

Solve for x. Express your answer in exact form.

2^{2x} - 6\cdot2^x - 16 =0

x=3

Solving Logarithmic Equations

200

Solve for x. Express your answers in exact form.

\log_6 (x+1) = 1 - \log_6 x

x=2

300

Solve for x. 

\log_5 x - 9 = \log_5 \sqrt{x}

Express your answer in the form, 

 x=b^y where  b, y \in\mathbb{Z} 

x=5^{18}

400

Solve for x. Express your answer in exact form.

\log_3 5 + 2 = \log_9 x

x=2025

500

Solve for x. Express your answer in exact form.

\log_2 x+\ln x = 1

x = e^{\frac{\ln 2}{1 + \ln 2}}

Exp and Log Properties

300

Write in terms of a single logarithm. 

 \log_2 3 + \log_4 3 

Express your answer in the form,  \log_b a^q , where  a\in\mathbb{Z}  and  q\in\mathbb{Q} 

\log_2 3^{\frac{3}{2}}

400

Write the following expression as a single logarithm in simplest form,

\frac{1}{2}\log_2 72 - \frac{3}{2}

\log_2 3

Applications

300

Consider an investment of $1000 at 3.5% annual interest rate, growing subject to compound interest for 10 years.

Show that the ratio of interest earned when compounded monthly compared against compounded quarterly is less than the ratio of interest earned when compounded quarterly compared against compounded annually.

\frac{1000(1+\frac{3.5}{1200})^120}{1000(1+\frac{3.5}{400})^40}<\frac{1000(1+\frac{3.5}{400})^40}{1000(1+\frac{3.5}{100})^10}