Logarithm Properties Worksheets Grade 11

1. Mason is studying a logarithmic spiral on a polar coordinate grid. The spiral is described by the equation r = 9 * 3^(θ/2), where r is the distance from the origin and θ is the angle in radians. Using properties of logarithms, rewrite the expression for log₃(r) as a linear function of θ. Then, determine the exact value of log₃(r) when θ = 8 radians.
2. log₂(x) + log₂(x - 7) = 3
3. A biologist is studying bacterial growth in a lab culture. The population P(t) after t hours is modeled by the exponential function P(t) = 5000 × e^(0.15t). Liam needs to determine how long it will take for the bacterial population to reach 25,000. Using properties of logarithms, find the time required for the population to reach this level.

…and 3 more problems

1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (8,0), and (8,6). A circle is circumscribed around this triangle, passing through all three vertices. What is the exact area of this circumscribed circle?
2. Aroha is an environmental scientist studying the acidity of rainwater in a forest. She measures the pH of a sample using the formula pH = -log[H⁺], where [H⁺] is the hydrogen ion concentration in moles per liter. She collects two samples: one from an area affected by pollution with a hydrogen ion concentration of 7 × 10⁻⁵ M, and another from a pristine area with a concentration of 1 × 10⁻⁶ M. Using properties of logarithms, determine how many times more acidic the polluted sample is compared to the pristine sample.
3. A right triangle is positioned on a coordinate plane with vertices at (0,0), (8,0), and (8,6). A circle is circumscribed around this triangle, passing through all three vertices. What is the exact area of this circumscribed circle?

…and 4 more problems

1. log₇(343x) + log₇(x/7) = 5
2. Dr. Rodriguez is analyzing sound intensity levels in a concert hall. The decibel level of a sound is given by the formula L = 10 log(I/I₀), where I is the sound intensity and I₀ is the reference intensity of 10⁻¹² W/m². During a rock concert, the sound intensity reaches 0.01 W/m². What is the sound intensity level in decibels?
3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (8,0), and (8,6). A circle is inscribed within this triangle, tangent to all three sides. What is the exact area of the circle? (Express your answer in terms of π)

…and 5 more problems