Exponential and Logarithmic Functions
1 answer:
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
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