Intensity Increase for 30dB & 22dB: Log Homework Solved

[carmenn]

Homework Statement

A hearing aid increases the sound intensity level, thereby allowing a person to hear better. For the following decibel increases, by how much does the intensity of sound increase?

a)30dB
b) 22dB

Homework Equations

$$\beta=10log (I/Io)$$

The Attempt at a Solution

I substitute 30 in for the beta, and 1x10^-12 for the Io, but i can't seem to solve for the correct answer. Can anyone lend a hand?

I really have trouble with bringng to log to the other side. Thanks

 

[Gib Z]

$$\frac{\beta}{10} = \log_{10} (I/I_0)$$

By the definition of the logarithm, $$I/I_0 = 10^{\frac{\beta}{10}}$$

So $$I = I_0 \cdot 10^{\frac{\beta}{10}}$$

 

[ogb p]

I would recommend using a scientific calculator to solve this problem. The equation you have provided is correct, but you need to use the inverse function of logarithm, which is 10^x, to solve for the intensity (I). The correct steps for solving this problem are:

1. Rearrange the equation to solve for I: I = 10^(β/10) * Io
2. Substitute the given values for β and Io: I = 10^(30/10) * 1x10^-12 and I = 10^(22/10) * 1x10^-12
3. Use a calculator to solve for I: I = 1x10^-6 and I = 1x10^-8
4. Convert the answers to scientific notation: I = 1μW and I = 100pW

Therefore, the intensity of sound increases by 1μW for a 30dB increase and by 100pW for a 22dB increase.