Complete the table below comparing exponential equations with the equivalent logarithmic equation. Then first row is done for you.

Question

Complete the table below comparing exponential equations with the equivalent logarithmic equation. Then first row is done for you.

b^x=y is the same as 〖log〗_b y=x
Exponential Logarithmic
2^3=8 log_2 8=3
5^2=25
10^4=10,000
log⁡1,000,000=x
e^x=12
ln⁡〖4=y〗

The population of a city (in thousands) closely fits the exponential function P(x)=500e^0.04x where x is the number of years since the year 2000.
Compute the expected population of the city in the year 2005.

After how many years would the population of the city reach 800 thousand?

Answer 1

Exponential Logarithmic

2^3=8 log_2 8=3
5^2=25 log_5 25=2
10^4=10,000 log_10 10,000=4
log_1,000,000=x log_1,000,000=x
e^x=12 ln 12=x
The population of a city (in thousands) closely fits the exponential function P(x)=500e^0.04x where x is the number of years since the year 2000.
Compute the expected population of the city in the year 2005:

To compute the expected population of the city in 2005, we substitute x = 5 (since it is the number of years since 2000) into the exponential equation P(x)=500e^0.04x.
P(5)=500e^0.04(5)
P(5)=500e^0.2
P(5)≈500(1.22140)
P(5)≈610.70
Therefore, the expected population of the city in the year 2005 is approximately 610,700.

After how many years would the population of the city reach 800 thousand:

To find the number of years it takes for the population to reach 800,000, we need to set up and solve the exponential equation P(x)=500e^0.04x=800.
500e^0.04x=800
e^0.04x=800/500
e^0.04x=1.6
Take the natural logarithm (ln) of both sides to solve for x:
ln(e^0.04x)=ln(1.6)
0.04x=ln(1.6)
x=(ln(1.6))/(0.04)
Using a calculator, we find that x is approximately 21.98.
Therefore, it would take approximately 22 years for the population of the city to reach 800,000.

Answer 2

Exponential Logarithmic

5^2=25 log_5 25=2
10^4=10,000 log_10 10,000=4
log⁡1,000,000=x log_1,000,000 x=6
e^x=12 ln 12=x

To compute the expected population of the city in the year 2005, we need to substitute x = 5 into the exponential equation P(x)=500e^0.04x:
P(5) = 500e^0.04(5)
P(5) = 500e^0.2
P(5) ≈ 500(1.219)
P(5) ≈ 609.5 thousand

Therefore, the expected population of the city in the year 2005 is approximately 609.5 thousand.

To find out after how many years the population of the city would reach 800 thousand, we need to solve the exponential equation P(x)=500e^0.04x for x:
800 = 500e^0.04x

Divide both sides of the equation by 500:
1.6 = e^0.04x

To isolate x, take the natural logarithm (ln) of both sides:
ln 1.6 = 0.04x

Now, divide both sides by 0.04:
x ≈ ln 1.6 / 0.04

Using a calculator, we find:
x ≈ 9.84

Therefore, the population of the city would reach 800 thousand after approximately 9.84 years.

Answer 3

To complete the table comparing exponential equations with their equivalent logarithmic equations, we need to find the values for the remaining equations.

For the equation 5^2=25, we can write the logarithmic form as log_5 25 = 2.

For the equation 10^4=10,000, we can write the logarithmic form as log_10 10,000 = 4.

For the equation log₁₀ 1,000,000 = x, we can find the value of x by evaluating the logarithm. 1,000,000 can be written as 10^6, so log₁₀ 1,000,000 = 6. Therefore, x = 6.

For the equation e^x = 12, we can write the logarithmic form as ln 12 = x, where ln is the natural logarithm.

For the equation ln 4 = y, there is no exponential equation given, so we can just leave it as ln 4 = y.

Now let's move on to the problem about the population of a city:

The exponential function for the population of the city is given as P(x) = 500e^0.04x, where x is the number of years since the year 2000.

To compute the expected population of the city in the year 2005, we substitute x = 5 (since 2005 - 2000 = 5) into the equation:

P(5) = 500e^(0.04*5)
P(5) = 500e^0.2
P(5) ≈ 500 * 1.22140 (approximating e^0.2 ≈ 1.22140 using a calculator)
P(5) ≈ 610.70

Therefore, the expected population of the city in the year 2005 is approximately 610,700.

To determine after how many years the population of the city would reach 800 thousand, we need to solve the equation P(x) = 800. We substitute P(x) = 800 and solve for x:

800 = 500e^(0.04x)

Dividing both sides by 500:
1.6 = e^(0.04x)

To isolate the exponential term, we take the natural logarithm of both sides:

ln 1.6 = ln (e^(0.04x))

Using the property ln e^a = a:
ln 1.6 = 0.04x

Now, we divide both sides by 0.04 to solve for x:
x = ln 1.6 / 0.04

Using a calculator, we find:
x ≈ 17.32

Therefore, it would take approximately 17.32 years (or rounded to 18 years) for the population of the city to reach 800 thousand.