2.12 Logarithmic function manipulation
State the product property for logarithms.
What does the product property for logarithms imply graphically?
State the power property for logarithms.
What does the power property for logarithms imply graphically?
State the change of base property for logarithms.
Apply the product property for logarithms to simplify the following expressions:
\text{log}_2(3x)
\text{log}_5 (7x)
\text{log}_3(5x)
\text{log}_4 (6x)
Apply the power property for logarithms to simplify the following expressions:
\text{log}_5 (x^4)
\text{log}_7 (x^2)
\text{log}_3 (x^6)
\text{log}_4 (x^3)
\text{log}_{10} (x^5)
\text{log}_2 (x^7)
Use the change of base property to rewrite the following expressions in terms of natural logarithms:
\text{log}_2 (8)
\text{log}_3 (18)
\text{log}_4 (16)
\text{log}_5 (20)
\text{log}_6 (12)
\text{log}_7 (49)
Suppose a logarithmic function experiences a horizontal dilation. Describe the equivalent vertical translation.
Suppose a logarithmic function has its input raised to a power. Describe the resulting vertical dilation.
Suppose you have the logarithmic function f(x) = \text{log}_2 (3x). Use the product property to write this function as a sum of two logarithms.
Suppose you have the logarithmic function f(x) = \text{log}_2 (x^3). Use the power property to simplify this function.
Suppose you have the logarithmic function f(x) = \text{log}_5 (x). Use the change of base property to rewrite this function using natural logarithms.
A certain population P is modeled by the function P(t) = 5000 \text{ log}_2 (t+1) where t is time in years. According to this model, how does the population change over time? What does this imply about the rate of growth of the population?
Sound intensity is measured in decibels using the formula \text{dB} = 10 \text{log}_{10} \left(\dfrac{I}{I_0}\right) where I is the intensity of the sound and I_0 is the reference intensity. Suppose you increase the intensity of a sound such that its decibel level increases by 20 \text{ dB}. By what factor did the intensity of the sound increase?
The Richter scale, which measures the magnitude of earthquakes, utilizes logarithms. An earthquake that measures 5.0 on the Richter scale is 10 times more powerful than an earthquake that measures 4.0. If an earthquake measures 7.0 on the Richter scale, how much more powerful is it than an earthquake that measures 5.0?