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Exponential and Logarithmic Model Calculator

Exponential and logarithmic functions are used to help answer question and make predictions about many different phenomena in the world and beyond.

INSTRUCTIONS

1. Which of the following axis of rotation would yield a representative cross-section of the solid as a disk?

X-axis Y-axis Both a and b None of the above

In 1798, the English cleric and scholar Reverend Thomas Robert Malthus stated that with unlimited resources, such as food and space, the population of a species grows geometrically. In modern notation, we say that the population P(t) of a species at time t is given by
P(t) = P₀e^rt,
where P₀ is the initial population, r is a growth constant, and e ≈ 2.71828 . . . is the base of the natural logarithms. Construct a model and affect the graph by adjusting the parameters P₀ and r, and investigate your model by setting values for t and P(t).
P₀ :	r :
t :	P(t) :

In the 1930’s, Charles Richter developed the Richter earthquake magnitude scale to quantify the energy released by an earthquake. He derived the formula

where R is the Richter scale number of the earthquake, A is the maximum deflection in millimeters (mm) of a seismograph needle at a distance of 100 km from the epicenter, A₀ = 0.001 mm is the maximum needle deflection caused by the smallest detectable earthquake, and the logarithm is the common (base 10) logarithm. Investigate the model by setting values for R and A.
A :	R :

Radioactive substances can be classified by their half-life, which is the time it takes for half of the substance to decay into a stable, non-radioactive form. For instance, strontium-90 has a half-life of 28.8 years, so a rock containing 100g of strontium-90 today will have only 50g of strontium-90 in 28.8 years. Mathematically, the amount A(t) of a radioactive substance remaining in a sample at time t is given by
A(t) = A₀2^-t/h,
where A₀ is the initial amount of the substance and h is the half-life. Construct a model and affect the graph by adjusting the parameters A₀ and h, and investigate your model by setting values for t and A(t).
A₀ :	h :
t :	A(t) :

You can tell the difference between total silence and a single voice, or between 1 voice or 2 voices, or even between 2 voices and 6. But can you tell the difference between 1000 voices and 1010? The diminishing ability to sense differences which are small compared to their source has been identified and quantified as the Weber-Fechner law. In the case of sound, it is expressed mathematically as

where L is the sound level in decibels, P is the sound power in Watts (W), P₀ = 10-12 W is a sound power reference value, and the logarithm is the common (base 10) logarithm. Investigate the model by setting values for P and L.
P :	L :

A Gaussian, or normal, distribution is a function used in probability to understand and make predictions. A question such as “How much of the data had a value below 63?” would be answered by the picture above. The bell curve is the graph of a normal distribution related to the average and variability of some data, and is formed by the equation.

where ¯x is the mean, or average, of the data, σ is the standard deviation of the data, e ≈ 2.718281828459 . . . is the base of the natural logarithms, and π ≈ 3.14159265358979 . . . . The answer to the question, 90.3%, is the proportion of area beneath the curve lying to the left of the vertical line x = 63. Construct a model and affect the graph by adjusting the parameters ¯x and σ and investigate your model by setting values for x.
¯x :	σ :
x :	area :

The strength of acids and bases are specified by their pH. A solution with a pH between 0 and 7 is an acid, with lower values corresponding to stronger acids. A solution with a pH between 7 and 14 is a base, with higher values corresponding to stronger bases. The pH of water, which is neutral, is 7. The pH of a substance is given by
pH = - log [H⁺] ,
where [H⁺] is the hydrogen ion concentration in moles/liter and the logarithm is the common (base 10) logarithm. Investigate the model by setting values for [H⁺] and pH.
[H⁺] :	pH :


Isaac Newton said the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings. We can say that the temperature T(t) of an object at time t is given by

where A is the ambient (room) temperature, T₀ is the initial temperature of the object, k is a cooling constant dependent on the material and environment, and e ≈ 2.718281828459 . . . is the base of the natural logarithms. Construct a model and affect the graph by adjusting the parameters A, T₀ and k, and investigate your model by setting values for t and T(t).
A :	T₀ :		k :
t :		T(t) :

It is easy to tell how many digits are in a whole number when it is written in standard decimal form: simply count the digits. But how many digits are in 57643⁶⁴⁸ ∙ (864975⁶³ / 245²⁷)³⁹? Logarithms and their properties offer a means to answering the question in cases like these. The number N of digits in the base-10 representation of a whole number n is given by

where represents the floor function, whose output is the largest whole number less than or equal to x, and the logarithm is the common (base 10) logarithm. Investigate the model by setting values for n and N.
n :	N :


In 1938, Brussels mathematician Pierre Verhulst improved Malthus’ population growth model. Assuming that a population’s surroundings can support only a limited number C of individuals, he proposed that a better model for a population P(t) at time t was given by the function

where k is a constant related to the growth rate, C is the carrying capacity of the environment, m is the time it takes the population to reach C/2, and e is the base of the natural logarithm. Construct a model and affect the graph by adjusting the parameters C, m, and k, and investigate your model by setting values for t and P(t).
C :	m :		k :
t :		P(t) :

Taking a hint from the ancient Greeks, astronomer N. R. Pogson came up with a way of classifying stars that extended the old numbering system into one that could accommodate modern astronomy. He defined the apparent magnitude m of a star as

where B is the observed brightness of the star, B₀ = 1 is the brightness of a hypothetical reference star, and the logarithm is the common (base 10) logarithm. Investigate the model by setting values for B and m.
B :	m :