Question

Exponential and Logarithmic Functions

Answer 1

Answer:

1. A. 999/2

2. B. 1%; exponential decay

3. C. 0.9962

4. C. logb(y⁵ * x⁶)

Step-by-step explanation:

1.

log(2x + 1) = 3

Solve Logarithm.

log(2x + 1) = 3

10^log(2x + 1) = 10^3(Take exponent of both sides)

2x + 1 = 10^3

2x + 1 = 1,000

2x + 1 − 1 = 1000 − 1 (Subtract 1 from both sides)

2x = 999

2x/2 = 999/2 (Divide both sides by 2)

x = 999/2

2.

y = (0.99)^x

Step 1: We compare the given function with the general form of exponent function

y = a (1 + r)^x

Step 2: We compare both equations and get:

1 + r = 0.99

1 - 1 + r = 0.99 - 1

r = -0.01

Step 3: We turn 'r' into a percentage

r = -0.01 × 100 = -1%

r = -1%

In the function f (x) = bx when b > 1, the function represents exponential growth.

In the function f (x) = bx when 0 < b < 1, the function represents exponential decay.

So since b = 0.99 in this equation, the function represents exponential decay

3. The number e , sometimes called the natural number, or Euler's number, is an important mathematical constant approximately equal to 2.7182822

3(2.7182822^2x) + 5 = 27

Step 1: Subtract -5 from both sides.

3(2.718282^2x) + 5 − 5 = 27 − 5

3(2.718282^2x) = 22

Step 2: Divide both sides by 3.

3(2.718282^2x)/3 = 22/3

the 3 in front of (2.718282^2x) cancels out when divided by 3

2.718282^2x = 22/3

log(2.718282^2x) = log(22/3) (Take log of both sides)

The logarithm of a number raised to a power is the power times the logarithm of the number.

2x * (log(2.718282) = log(22/3)

2x = log(22/3) / log(2.718282)

log(22/3) is equal to 0.86530142 in decimal form and the Log base 10 of 2.718282 is approximately 0.4342945

so the equation would be 0.86530142/0.4342945 = 1.99243

2x = 1.99243

2x = 1.99243 (Divide both sides by 2)

2x/2 = 1.99243/2

x = 0.996215 = 0.9962 rounded to the nearest thousandth.

4.  Steps in picture so you understand better