Task Sets: Express exponential solutions as logarithms - Common Core Math - HS Functions

A

Warm-Up: Review What You Know

These problems review skills from earlier lessons.

1.

Which of the following is equivalent to $2^5 = 32$ ?

A.

$\log_{32} 2 = 5$

B.

$\log_2 32 = 5$

C.

$\log_5 32 = 2$

D.

$\log_2 5 = 32$

2.

To solve $5 \cdot 3^t = 135$ , the first step is to isolate $3^t$ . What is the result of that first step?

A.

$3^t = 27$

B.

$3^t = 130$

C.

$\log_3(3^t) = \log_3(135)$

D.

$3^t = 675$

3.

Which statement correctly describes what $\log_{10} 1000$ equals and why?

A.

3, because $10^3 = 1000$

B.

100, because $\log_{10} 1000 = 1000 \div 10$

C.

4, because $\log_{10} 1000 = \log_{10} 10 \times \log_{10} 100$

D.

30, because $\log_{10} 1000 = 10 \times 3$

B

Fluency Practice

1.

Solve $4 \cdot 2^t = 64$ . Write $t$ as a logarithm and then evaluate it.

A.

$t = \log_2 16 = 4$

B.

$t = \log_2 64 = 6$

C.

$t = \log_2 16 = 3$

D.

$t = \log_4 64 = 3$

2.

Solve $3 \cdot 10^t = 300$ . Write $t$ as a base-10 logarithm and evaluate. What is $t$ ?

3.

Solve $200 \cdot 10^{0.5t} = 5000$ . Round your answer to two decimal places.

4.

Solve $50 \cdot e^{0.04t} = 150$ . Round your answer to two decimal places.

5.

Solve $12 \cdot 2^{t/3} = 192$ . Round your answer to two decimal places if needed.

C

Varied Practice

D

Word Problems

1.

A city's population is modeled by $P = 80000 \cdot 10^{0.02t}$ , where $t$ is years after 2010.

In what year will the population reach 200,000? Round to the nearest whole year.

2.

An investment account grows continuously at 5% per year. The balance after $t$ years is modeled by $A = 1500 \cdot e^{0.05t}$ .

1.

How many years does it take for the balance to reach $4000? Round to two decimal places.

2.

Approximately how much longer does it take for the balance to go from $4000 to $6000? Round to the nearest year.

3.

A radioactive substance has a decay model $A = 500 \cdot 2^{-t/8}$ , where $A$ is the amount in grams and $t$ is time in years.

After how many years will only 50 grams remain? Round to two decimal places.

4.

A bacterial culture starts with 400 cells and doubles every 5 hours. The model is $N = 400 \cdot 2^{t/5}$ .

After how many hours will there be 12,800 cells? Show all three steps.

5.

An investment of $P$ dollars grows continuously at 8% per year. The doubling-time formula gives $t = \frac{\ln(2)}{0.08}$ .

Compute the doubling time to two decimal places. Then identify what is special about this answer — why does the initial value $P$ not matter?

E

Error Analysis

1.

Kenji solved $500 \cdot 2^{0.1t} = 4000$ :

$\log(500 \cdot 2^{0.1t}) = \log(4000)$
$\log(500) + 0.1t \cdot \log(2) = \log(4000)$
$0.1t \cdot \log(2) = \log(4000) - \log(500)$
$0.1t = \frac{\log(8)}{\ \log(2)} = 3$
$t = 30$

Kenji got $t = 30$ . Is this correct? If not, identify the error.

A.

Correct — Kenji's answer is right.

B.

Incorrect — Kenji should have divided by 500 first before taking the log.

C.

Incorrect — Kenji used $\log$ instead of $\ln$ ; the answer should use the natural log.

D.

Incorrect — In the last step, Kenji divided by $\log(2)$ but should have subtracted it.

2.

Priya solved $10 \cdot e^{0.2t} = 80$ :

Step 1: $e^{0.2t} = 8$

Step 2: $0.2t = \log(8)$

Step 3: $t = \frac{\log(8)}{0.2} \approx \frac{0.903}{0.2} \approx 4.52$

What error did Priya make in Step 2?

A.

Priya used $\log$ (base 10) instead of $\ln$ (base $e$ ), so Step 2 should read $0.2t = \ln(8)$ .

B.

Priya wrote $\log(8)$ but should have written $\log(8)/\log(e)$ , using the change-of-base formula.

C.

Priya correctly isolated in Step 1, but Step 2 should be $0.2t = \log_8(e)$ because we're solving for the exponent on $e$ .

D.

Priya forgot to divide by 10 before step 2; the equation should be $e^{0.2t} = 70$ first.

F

Challenge / Extension

1.

A population is modeled by $P = 2000 \cdot e^{0.06t}$ and separately by $Q = 5000 \cdot 10^{0.02t}$ , where $t$ is years from now.

After how many years will the two populations be equal? Round to two decimal places.

2.