log_base 3 1/27. Evaluate the log. Show all steps
To evaluate log_base 3 1/27, we first need to express 1/27 as a power of 3:
1/27 = 3^(-3)
Now, we can rewrite the log expression using the power:
log_base 3 3^(-3)
Since the base and the argument are the same, we can simplify this to:
-3
Therefore, log_base 3 1/27 = -3.
log_base 3 1/27. Evaluate the log
To evaluate the log_base 3 1/27, we first need to express 1/27 as a power of 3:
1/27 = 3^(-3)
Now, we can rewrite the log expression using the power:
log_base 3 3^(-3)
Since the base and the argument are the same, we can simplify this to:
-3
Therefore, log_base 3 1/27 = -3.
n 4/3 = m. Write the equation in log form. Show all steps.
To write the equation n 4/3 = m in log form, we first need to understand that logarithms are the inverse operation of exponentiation. The equation n 4/3 = m can be written in exponential form as:
3^(4n) = m
Now we can write this exponential equation in logarithmic form:
log_base 3 m = 4n
Therefore, the equation n 4/3 = m in log form is log_base 3 m = 4n.
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A coordinate graph is shown. The horizontal axis extends from 0 to 12 years. The vertical axis extends from 0 to 9500 with an axis label of 'Value' in dollars. A curve is graphed which begins at left parenthesis 0 comma 3500 right parenthesis, then decreases passing through approximately left parenthesis 1 comma 2700 right parenthesis.
The exponential decay graph shows the expected depreciation for a new boat, selling for $3,500, over 10 years.
Write an exponential function for the graph. Use the function to find the value of the boat after 9.5 years.
To find the exponential function for the graph, we need to use the general form of an exponential decay function:
f(x) = a * (1 - r)^x
where:
- a is the initial value ($3,500 in this case)
- r is the decay rate
- x is the time in years
Since we know that the boat starts at $3,500 and decreases to approximately $2,700 after 1 year, we can write this information as:
$2,700 = $3,500 * (1 - r)^1
Dividing both sides by $3,500, we get:
0.77 = 1 - r
r = 1 - 0.77
r = 0.23
So, the exponential function for the graph is:
f(x) = 3500 * (1 - 0.23)^x
f(x) = 3500 * 0.77^x
To find the value of the boat after 9.5 years, we substitute x = 9.5 into the function:
f(9.5) = 3500 * 0.77^9.5
f(9.5) ≈ 3500 * 0.1698
f(9.5) ≈ $593.30
Therefore, the value of the boat after 9.5 years is approximately $593.30.
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In a particular region of a national park, there are currently 330 deer, and the population is increasing at an annual rate of 11%.
Write an exponential function to model the deer population.
Explain what each value in the model represents.
Predict the number of deer that will be in the region after five years. Show your work.
To model the deer population using an exponential function, we can use the growth formula:
P(t) = P₀ * (1 + r)^t
where:
- P(t) is the population after t years
- P₀ is the initial population (330 deer in this case)
- r is the annual growth rate (11% in this case)
- t is the time in years
Given that the initial population P₀ is 330 deer and the growth rate r is 11% (0.11 in decimal form), the exponential function for the deer population is:
P(t) = 330 * (1 + 0.11)^t
P(t) = 330 * 1.11^t
Each value in the model represents the following:
- P(t) is the predicted population of deer after t years
- 330 is the initial population of deer
- 1.11 is the growth factor that corresponds to an annual growth rate of 11%
- t is the time in years
To predict the number of deer that will be in the region after five years, we substitute t = 5 into the exponential function:
P(5) = 330 * 1.11^5
P(5) = 330 * 1.77156
P(5) ≈ 585.1
Therefore, the predicted number of deer in the region after five years is approximately 585 deer.