# Logarithim question decreasing in value

• pinnacleprouk
In summary, the conversation discusses the modeling of radioactivity using the equation R = A x 2 ^ -Bt, where R is measured in becquerels per gram, t is measured in hours, and A and B are constants. The group then solves for A and B, and calculates the time for the radioactivity to reduce to half its initial value (half life). The correct value for B is 0.2630 and the half life is approximately 3.8 hours.
pinnacleprouk
Logarithim question "decreasing in value"

## Homework Statement

The Radioactivity (R) of a substance can be modeled using the equation:

R = A x 2 ^ -Bt

Where R is measured in becquerels per gram, t is measured in hours and A and B are constants.
If a substance has an initial radioactivity of 60 becquerels and one hour later it is 50 becquerels find A and B and the time for the radioactivity to reduce to half its initial value (Half Life)

R = A x 2 ^ -Bt

## The Attempt at a Solution

R = A x 2 ^ - Bt

R = 60 t = 0
R = 50 t = 1
60 = A x 2 ^ - Bx0

60 = A

50 = 60 x 2 ^ - B

2 - B = 50/60

2 - B = 0.8

log 2 - B = log 0.8

- B log 2 = log 0.8

- B = log 0.8/log 2

- B = - 0.32

Is this all right?

I work out half life to be around 3 hours but could anybody give me a more exact time?

Solved t = 3.125

Thanks

pinnacleprouk said:

## Homework Statement

The Radioactivity (R) of a substance can be modeled using the equation:

R = A x 2 ^ -Bt

Where R is measured in becquerels per gram, t is measured in hours and A and B are constants.
If a substance has an initial radioactivity of 60 becquerels and one hour later it is 50 becquerels find A and B and the time for the radioactivity to reduce to half its initial value (Half Life)

R = A x 2 ^ -Bt

## The Attempt at a Solution

R = A x 2 ^ - Bt

R = 60 t = 0
R = 50 t = 1
60 = A x 2 ^ - Bx0

60 = A

50 = 60 x 2 ^ - B
The next line is incorrect. It should be 2-B = 50/60 = 5/6
pinnacleprouk said:
2 - B = 50/60
The next line is also incorrect. 5/6 != .8
pinnacleprouk said:
2 - B = 0.8

log 2 - B = log 0.8

- B log 2 = log 0.8
The next line is incorrect, but at least I understand what you are doing. It would be correct if you had on the right side (log 5/6)/log 2
pinnacleprouk said:
- B = log 0.8/log 2

- B = - 0.32
You're actually off by quite a bit. To 4 decimal places, B = .2630
pinnacleprouk said:
Is this all right?

I work out half life to be around 3 hours but could anybody give me a more exact time?
pinnacleprouk said:

## 1. What is a logarithm?

A logarithm is the inverse function of exponentiation. It is used to determine the power to which a base number must be raised to equal a given number. For example, the logarithm of 100 to the base 10 is 2, because 10 raised to the power of 2 equals 100.

## 2. How does a logarithm decrease in value?

A logarithm decreases in value as the base number increases. For example, the logarithm of 100 to the base 10 is 2, but the logarithm of 100 to the base 1000 is 0.1. This is because a larger base number requires a smaller power to equal the same number.

## 3. What is the relationship between logarithms and exponential functions?

Logarithms and exponential functions are inverse operations, meaning they undo each other. The logarithm of a number to a certain base is equal to the exponent of that base. For example, log28 = 3 and 23 = 8.

## 4. How are logarithms used in real life?

Logarithms are used in a variety of fields, including science, engineering, finance, and statistics. In science, they are used to measure the intensity of earthquakes and sound, as well as the acidity of a substance. In finance, they are used to calculate compound interest. In statistics, they are used to transform data to make it more easily interpreted.

## 5. Can a logarithm ever be negative?

Yes, a logarithm can be negative. This occurs when the base is greater than 1 and the number being evaluated is between 0 and 1. For example, log20.5 = -1, because 2 raised to the power of -1 equals 0.5.

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