1. A simple exponential equation is given below.
`e^(2x) + 3 = 5 `
The answer can be given if needed. But the basic property required to solve this equation is given by,
`log_b b^x = x ` , in the above example the logarithm used is natural logarithm.
2. One of the most widely used logarithmic equations is the "decibel equation" which is used to express the sound intensity above the standard threshold of hearing. This method was introduced by the people at "Bell Telephone Laboratories" to measure the signal losses in telephone circuits and named as "decibel" to honor Alexander Graham Bell.
The equation is given by,
`I(dB) = 10log_10(I/I_0) `
The logarithmic base in this equation is 10.
Where, `I(dB)` is the sound intensity in decibels. I is the actual sound intensity in `Wm^-2` at a certain point in the path of the sound propagation. `I_0` is the reference sound intensity or otherwise known as the threshold of hearing. `I_0` for human ear is `10^-12 Wm^-2` .
Ex: Imagine at a certain location, we hear a sound which has an intensity of 1000 times of the threshold of hearing (I_0). The sound level in decibels can be given by,
`I(dB) = 10log_10((1000I_0)/I_0)`
`I(dB) = 10log_10(1000)`
`I(dB) = 10log_10(10^3)`
`I(dB) = 10*3 = 30`
The sound level is 30dB.