Question 1189074
the formula given is:


m = log (a / 10^-6) where a is the amplitude and m is the magnitude.


if the amplitude is 10^-3, then the formula becomes:


m = log(10^-3 / 10^-6) = log(10^-3+6)) = log(10^3) = 3


if m = 7.2, then 7.2 = log(a / 10^-6)


this becomes 7.2 = log(a) - log(10^-6) which becomes 7.2 = log(a) - -6 * log(10) which becomes 7.2 = log(a) + 6 * log(10) which becomes 7.2 = log(a) + 6.


subtract 6 from both sides of the eqution to get:


7.2 - 6 = log(a)


simplify to get:


1.2 = log(a)


this is true if and only if 10^1.2 = a


sove for a to get:


a = 15.84893192.


that should be the amplitude.


confirm by replacing a in the original equation by that to get:


7.2 = log(a / 10^-6) becomes 7.2 = log(15.84893192 / 10^-6) = 7.2.


this confirms the solution is correct.


some log properties that were used.


log(a/b_)  log(a) - log(b)


y = log(x) if and only if 10^y = x


and:


10^x = y if and only if log(y) = x


log(x) means the same as log10(x).


log10(x) means the log of b to the base of 10.


log10 is the log function on your calculator.


the general equation is:


loga(b) = c if and only if a^c = b


in reverse, a^c = b if and only if loga(b) = c


there is also a log base conversion formula that says.


loga(x) = logb(x) / logb(a)


this is useful to convert logs of any base to the base of 10 which is the log function of your calculator.


for example:


log2(16) = y if and only if 2^y = 16
solve for y to get y = 4 because 2^4 = 16.


using the base conversion formula, you could also have done:
log2(16) = log(16)/log(2) = 4 by using the log function of your calculator.


let me know if you have any questions.


theo


here's some references.


<a href = "https://www.chilimath.com/lessons/advanced-algebra/logarithm-rules/" target = "_blank">https://www.chilimath.com/lessons/advanced-algebra/logarithm-rules/</a>


<a href = "https://www.storyofmathematics.com/logarithm-rules" target = _blank">https://www.storyofmathematics.com/logarithm-rules</a>